General Topology
Proceedings of General Topology Symposium
held in Kobe, December 18 – 20, 2002

平成１４年科学研究費基盤研究（Ａ）
位相幾何学の総合的研究（課題番号 12304003）
代表者： 東京大学 坪井 俊

Contents
1. Keiko Chiba, Yukinobu Yajima,
Covering properties of inverse limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2. Naotsugu Chinen,
Topological sequence entropy of monotone maps on onedimensional continua . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3. Dikran Dikranjan , Dmitri Shakhmatov,
Hereditarily separable group topologies on Abelian groups . . . . . . . . . . . . . 13
4. Sakaé Fuchino,
On the property SEP of partial orderings . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5. V. Gutev, Haruto Ohta, Kaori Yamazaki,
Selections and sandwich-like properties via semi-continuous
Banach-valued funcitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35
6. Brendle Joerg,
The Baire space ordered by eventual domination: spectra . . . . . . . . . . . . . 45
7. Yukinobu Kemoto,
Products of 2 spaces and products of 3 spaces . . . . . . . . . . . . . . . . . . . . . . . . . 55
8. Masayuki Kurihara, Katsuro Sakai, Masato Yaguchi,
Hyperspaces with the Hausdoarff metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
9. Takahisa Miyata, Tadashi Watanabe,
Approximate resolutions and an application to Hausdorff
dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
10. Makoto Takahashi,
On strongly σ-short Boolean algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
11. Tatsuhiko Yagasaki,
Smoothness of scaling functions and transfer operators . . . . . . . . . . . . . . . .79
12. Iwao Yoshioka,
Closed images of spaces having g-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
ParticipantsParticipants (in Japanese)

Proceedings of General Topology Symposium, p. 1 - p. 7, 2002

http://gt2002.h.kobe-u.ac.jp/

COVERING PROPERTIES OF INVERSE LIMITS, II

Keiko Chiba and Yukinobu Yajima
Throughout this report, all spaces are topological spaces without any separation
axiom, and all maps are continuous. For an inverse system {Xα , πβα , Λ} and its
limit X, let Λ be a directed set with an order < and its cardinality λ, where λ ≥ ω,and let πα be the projection from X into Xα for each α ∈ Λ.
1. Known results and Qusetions
The following result of Bešlagić is a motivation of the study for the covering
properties of inverse limits.
Q
Theorem
1.1 [Be] (see [C1]). Let X = α∈Γ Xα be a product space such thatQ
α∈F Xα is normal for each finite F ⊂ Γ. Then X is normal if and only if it isλ-paracompact, where λ is the cardinality of Γ.
However, the “if” part of Theorem 1.1 had been already extended by Aoki:
Theorem 1.2 [A]. Let {Xα , πβα , Λ} be an inverse system and X its inverse limit
with each projection πα being a pseudo-open map. Suppose that X is λ-paracompact.
If each Xα is normal, then so is X.
These results lead us to consider the following general statement:
Statement (∗). Let P be a topological property. Let {Xα , πβα , Λ} be an inverse
system and X its inverse limit with each projection πα being a pseudo-open map.
Suppose that X is λ-paracompact. If each Xα is P, then so is X.
Aoki and Chiba proved many results for P being several covering properties andsome other separation properties in the Statement (∗) as follows.
Theorem 1.3 [A, C2, C4]. Let {Xα , πβα , Λ} be an inverse system and X its inverse
limit with each projection πα being a pseudo-open map. Suppose that X is λparacompact. If each Xα satisfies one of the following properties, then X has thecorresponding property.
(1) Paracompactness.
(2) Collectionwise normality.
(3) Subparacompactness.
(4) Metacompactness.
(5) Submetacompactness (= θ-refinability).
(6) Subnormality.
Moreover, we can consider another general statement:
Typeset by AMS-TEX

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KEIKO CHIBA AND YUKINOBU YAJIMA

Statement (∗∗). Let P be a topological property. Let {Xα , πβα , Λ} be an inverse
system and X its inverse limit. Suppose that X is hereditarily λ-paracompact. If
each Xα is hereditarily P, then so is X.
Chiba also proved a similar result to Theorem 1.3 for the Statement (∗∗) asfollows.
Theorem 1.4 [C2, C4]. Let {Xα , πβα , Λ} be an inverse system and X its inverse
limit. Suppose that X is hereditarily λ-paracompact. If each Xα satisfies one of thefollowing properties, then X has the corresponding property.
(0) Hereditary normality.
(1) Hereditary paracompactness.
(2) Hereditary collectionwise normality.
(3) Hereditary subparacompactness.
(4) Hereditary metacompactness.
(5) Hereditary submetacompactness (= hereditary θ-refinability).
(6) Hereditary subnormality.
The purpose of this study is to prove that Theorems 1.3 and 1.4 hold for all main
covering properties and all main separation properties. From this point of view, it
is natural to raise the following three questions:
Question 1 [C4]. (i) Does Theorem 1.3 hold for δθ-refinability?
(ii) Does Theorem 1.4 hold for hereditary δθ-refinability?
Question 2 [C4]. (i) Does Theorem 1.3 hold for collectionwise δ-normality?
(ii) Does Theorem 1.4 hold for hereditary collectionwise δ-normality?
Question 3 [C4]. (i) Does Theorem 1.3 hold for collectionwise subnormality?
(ii) Does Theorem 1.4 hold for hereditary collectionwise subnormality?
As a partial answer of Question 1 (i), Chiba obtained
Theorem 1.5 [C3, C4]. Let {Xα , πβα , Λ} be an inverse system and X its inverse
limit with each projection πα being a pseudo-open map. Suppose that X is λparacompact.
(1) If each Xα is normal and δθ-refinable, then X is δθ-refinable.
(2) If each Xα is δθ-refinable and Λ is countable, then X is δθ-refinable.
Remark. It should be noted that a certain assumption of X such as λ-paracompactness seems to be always necessary to consider the covering properties of inverse
limits. In fact, the product ω ω1 of uncountably many copies of ω (= the countableinfinite discrete space) is the limit of an inverse system of discrete spaces witheach projection being open. However, ω ω1 is not countably paracompact and not
subnormal and it is not even weakly δθ-refinable (see [CGP, 11.4]).
2. δθ-refinability and weak θ-refinability
A map f from X onto Y is pseudo-open if y ∈ Int f (U ) holds for each y ∈ Y andeach open set U in X with f −1 (y) ⊂ U . Note that both open and onto maps andclosed and onto maps are pseudo-open.
A space X is λ-paracompact if every open cover of X with cardinality ≤ λ has
a locally finite open refinement. A cover A of a space X is directed if for any
A0 , A1 ∈ A, there is A2 ∈ A with A0 ∪ A1 ⊂ A2 .

COVERING PROPERTIES OF INVERSE LIMITS, II

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Lemma 2.1 [M]. A space X is λ-paracompact if and only if for every directed open
cover U of X with cardinality ≤ λ, there is a locally finite open cover V of X suchthat {V : V ∈ V} refines U.
Recall that a space X is weakly θ-refinable (respectively, weaklyδθ-refinable)S
if for every open cover U of X, there is an open refinement n∈ω Vn of U suchthat for each x ∈ X one can find nx ∈ ω with ord(x, Vnx ) = 1 (respectively,
1 ≤ ord(x, Vnx ) ≤ ω).
Before we consider the Qustion 1 (i), we should also consider the following similar
question:
Question 10 . Does Theorem 1.3 hold for weak θ-refinability or weak δθ-refinability?
First, we can obtain an affirmative answer to this question as follows.
Theorem 2.2. Let {Xα , πβα , Λ} be an inverse system and X its inverse limit witheach projection πα being a pseudo-open map. Suppose that X is λ-paracompact. If
each Xα is weakly θ-refinable (weakly δθ-refinable), then so is X.
Next, we proceed to consider the Question 1 (i). For that, the following concept
plays an important role.
A space X is λ-subparacompact if every open cover of X with cardinality ≤ λ hasa σ-locally finite closed refinement. Let X be a space and V a collection of subsets
in X. For each x ∈ X, we denote by ord(x, V) the cardinality of {V ∈ V : x ∈ V }.
Lemma 2.3 [B1, B2]. For a space X, the following are equivalent.
(a) X is λ-subparacompact.
(b) Every open cover of X with cardinality ≤ λ has a σ-discrete closed refinement.
(c) Every open cover of X with cardinality ≤ λ has a σ-closure-preserving closed
refinement.
(d) For every open cover U of X with cardinality ≤ λ, there is a sequence {Vn }of open refinements of U such that for each x ∈ X one can find nx ∈ ω withord(x, Vnx ) = 1.
Remark. As is well-known, paracompactness implies subparacompactness. However, for each λ ≥ ω, λ-paracompactness does not imply λ-subparacompactness. In
fact, let Xλ = λ+ × (λ+ + 1). Since Xλ is the product of a λ-paracompact space
and a compact space, it is λ-paracompact. However, Xλ is not subnormal (because,by the pressing down lemma, {(α, α) ∈ Xλ : α ∈ λ+ } and λ+ × {λ+ } cannot be
separated by disjoint Gδ -sets). Since every ω-subparacompact space is subnormal,
Xλ is not ω-subparacompact, hence not λ-subparacompact.
A space X is subnormal if for any disjoint closed sets A and B in X, there
are disjoint Gδ -sets G and H such that A ⊂ B and B ⊂ H. Note that X is
subnormal if and only if every finite (or binary) open cover of X has a countableclosed refinement.
Lemma 2.4. Every λ-paracompact and subnormal space is λ-subparacompact.
A space X is δθ-refinable (= submetaLindelöf ) if for every open cover U of X,
there is a sequence {Vn } of open refinements of U such that for each x ∈ X one canfind nx ∈ ω with ord(x, Vnx ) ≤ ω.

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KEIKO CHIBA AND YUKINOBU YAJIMA

Using above lemmas, we can obtain a partial answer to the Question 1 (i). Thisis also a generalization of [C4, Theorem 1 (i)].
Theorem 2.5. Let {Xα , πβα , Λ} be an inverse system and X its inverse limit witheach projection πα being a pseudo-open map. Suppose that X is λ-paracompact andsubnormal. If each Xα is δθ-refinable, then so is X.
Theorems 1.3 (6) and 2.6 immediately yield
Corollary 2.6. Let {Xα , πβα , Λ} be an inverse system and X its inverse limit witheach projection πα being a pseudo-open map. Suppose that X is λ-paracompact. If
each Xα is δθ-refinable and subnormal, then so is X.
3. Collectionwise δ-normality and collectionwise subnormality
A space X is collectionwise δ-normal if for every discrete collection {Cξ : ξ ∈ Ξ}of subsets in X, there is a disjoint collection {Uξ : ξ ∈ Ξ} of Gδ -sets in X suchthat Cξ ⊂ Uξ for each ξ ∈ Ξ. It is clear that every collectionwise δ-normal space issubnormal.
The following is an affirmative answer to the Question 2 (i) (= [C4, Question
3]).
Theorem 3.1. Let {Xα , πβα , Λ} be an inverse system and X its inverse limit witheach projection πα being a pseudo-open map. Suppose that X is λ-paracompact. If
each Xα is collectionwise δ-normal, then so is X.
A space X is collectionwise subnormal (= discretely subexpandable) if for every
discrete collection {Cξ : ξ ∈ Ξ} of subsets in X, there is a sequence Un = {Uξ,n : ξ ∈Ξ}, n ∈ ω, of collections of open sets in X such that Cξ ⊂ Uξ,n for each ξ ∈ Ξ andeach n ∈ ω, and for each x ∈ X, one can find nx ∈ ω with ord(x, Un ) ≤ 1.
Note that every subparacompact space is collectionwise subnormal and that every collectionwise subnormal space is collectionwise δ-normal.
The following is an affirmative answer to Question 3 (i) (= [C4, Question 2]).
Theorem 3.2. Let {Xα , πβα , Λ} be an inverse system and X its inverse limit witheach projection πα being a pseudo-open map. Suppose that X is λ-paracompact. If
each Xα is collectionwise subnormal, then so is X.
Remark. Recall that a space X is finitely subparacompact if every finite open coverof X has a σ-discrete closed refinement, and that a space X is boundedly subexpandable if X is collectionwise subnormal (= discretely subexpandable) and finitely
subparacompact (see [K]). However, note that finite subparacompactness is equivalent to subnormality, and that collectionwise subnormality implies subnormality.
Hence collectionwise subnormality is exactly equivalent to bounded subexpandability. So [C4, Theorem 1 (vii)] would be an affirmative answer to our Question 3 (=[C4, Question 2]) if the proof would be correct. However, there is a gap in the proofof [C4, Theorem 1 (vii)] (more precisely, the part of “Proof of (1)” is not correct).
Consequently, we can consider our proof of Theorem 3.2, which is omitted here, asa correct one of [C4, Theorem 1 (vii)].
4. Hereditarily subnormality and related properties
Recall that a space X is hereditarily subnormal if every subspace of X is subnormal. Note that X is hereditarily subnormal if and only if every open subspaceof X is subnormal.

COVERING PROPERTIES OF INVERSE LIMITS, II

5

The former part of the following was actually stated in [C4] without proof.
Proposition 4.1 [C4]. Let {Xα , πβα , Λ} be an inverse system and X its inverse
limit. Let G be an open set of X. Suppose that G is either λ-paracompact or
λ-subparacompact. If each Xα is hereditarily subnormal, then G is subnormal.
Recall that a space X is hereditarily δθ-refinable if every (open) subspace of X
is δθ-refinable.
The following is a partial answer to Question 1 (ii).
Theorem 4.2. Let {Xα , πβα , Λ} be an inverse system and X its inverse limit. Let
G be λ-subparacompact open subspace of X. If each Xα is hereditarily δθ-refinable,
then G is δθ-refinable.
We also obtain a generalization of [C3, Theorem 2] as follows.
Proposition 4.3. Let {Xn , πkn , ω} be an inverse sequence and X its inverse limit.
Let G be a countably metacompact open subspace of X. If each Xn is hereditarilyδθ-refinable, then G is δθ-refinable.
A space X is hereditarily collectionwise δ-normal if every subspace of X is collectionwise δ-normal.
Theorem 4.4. Let {Xα , πβα , Λ} be an inverse system and X its inverse limit. Let G
be a λ-subparacompact open subspace of X. If each Xα is hereditarily collectionwise
δ-normal, then G is collectionwise δ-normal.
Note that a space X is hereditarily collectionwise δ-normal if and only if every
open subspace of X is collectionwise δ-normal. Thus Lemma 2.4, Proposition 4.1
and Theorem 4.4 immediately yield an affirmative answer to Question 2 (ii) (= [C4,Question 6]):
Corollary 4.5. Let {Xα , πβα , Λ} be an inverse system and X its inverse limit. Suppose that X is hereditarily λ-paracompact. If each Xα is hereditarily collectionwise
δ-normal, then so is X.
Recall that a space X is hereditarily collectionwise subnormal if every subspaceof X is collectionwise subnormal.
The following gives an affirmative answer to Question 3 (ii).
Theorem 4.6. Let {Xα , πβα , Λ} be an inverse system and X its inverse limit. Let G
be a λ-subparacompact open subspace of X. If each Xα is hereditarily collectionwise
subnormal, then G is collectionwise subnormal.
Since a space X is hereditarily collectionwise subnormal if and only if every
open subspace of X is collectionwise subnormal, Lemma 2.4, Proposition 4.1 andTheorem 4.6 immediately yield an affirmative answer to Question 3 (ii) (= [C4,Question 5]):Corollary 4.7. Let {Xα , πβα , Λ} be an inverse system and X its inverse limit. Suppose that X is hereditarily λ-paracompact. If each Xα is hereditarily collectionwise
subnormal, then so is X.

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KEIKO CHIBA AND YUKINOBU YAJIMA

5. Hereditarily subparacompactness
and hereditarily submetacompactness
We begin with the hereditary subparacompact case:
Proposition 5.1. Let {Xα , πβα , Λ} be an inverse system and X its inverse limit.
Let G be a λ-subparacompact open subspace of X. If each Xα is hereditarily subparacompact, then G is subparacompact.
For the hereditary submetacompact case, we need the following lemma.
Lemma 5.2 [GY]. There is a filter F on ω satisfying: For every submetacompact
space X and every open cover U of X, there is a sequence {Vn } of open refinementsof U such that for each x ∈ X, {n ∈ ω : ord(x, Vn ) < ω} ∈ F.
A space X is λ-submetacompact if for every open cover U of X with cardinality≤ λ, there is a sequence {Vn } of open refinements of U such that for each x ∈ X
one can find nx ∈ ω with ord(x, Vnx ) < ω. Clearly, λ-subparacompactness implies
λ-submetacompactness.
Theorem 5.3. Let {Xα , πβα , Λ} be an inverse system and X its inverse limit. Let
G be a λ-submetacompact open subspace of X. If each Xα is hereditarily submetacompact, then G is submetacompact.
A space X is λ-metacompact if every open cover of X with cardinality ≤ λ has apoint-finite open refinement. Clearly, λ-metacompactness implies λ-submetacompactness.
Proposition 5.4. Let {Xα , πβα , Λ} be an inverse system and X its inverse limit.
Let G be a λ-metacompact open subspace of X. If each Xα is hereditarily metacompact, then G is metacompact.
A space X is λ-weakly θ-refinable
S if for every open cover U of X with cardinality≤ λ, there is an open refinement n∈ω Vn of U such that for each x ∈ X one canfind nx ∈ ω with ord(x, Vnx ) = 1. Similarly, we have
Proposition 5.5. Let {Xα , πβα , Λ} be an inverse system and X its inverse limit.
Let G be a λ-weakly θ-refinable open subspace of X. If each Xα is hereditarily
weakly θ-refinable, then G is weakly θ-refinable.
References[A]

Y. Aoki, Orthocompactness of inverse limits and products, Tsukuba J. Math. 4 (1980),
241–255.
[Be]
A. Bešlagić, Normality in products, Topology Appl. 22 (1986), 71–82.
[B1]
D. K. Burke, On subparacompact spaces, Proc. Amer. Math. Soc. 23 (1969), 655–663.
[B2]
D. K. Burke, On p-spaces and w∆-spaces, Pacific J. Math. 35 (1970), 285–296.
[CGP] J. Chaber, G. Gruenhage and R. Pol, On Souslin sets and embeddings in integer-valued
function spaces on ω1 , Topology and Appl. 82 (1998), 71–104.
[C1]
K. Chiba, A remark on the normality of infinite products, Proc. Amer. Math. Soc. 105(1989), 510–512.
[C2]
K. Chiba, Normality of inverse limits, Math. Japonica 35 (1990), 959–970.
[C3]
K. Chiba, δθ-refinability in products, Q & A in Gen. Top. 10 (1992), 173–179.
[C4]
K. Chiba, Covering properties of inverse limits, Q & A in Gen. Top. 20 (2002) (to appear).
[GY] G. Gruenhage and Y. Yajima, A filter property of submetacompactness and its application
to products, Topology and Appl. 36 (1990), 43–55.
[K]
Y. Katuta, Expandability and its generalizations, Fund. Math. 87 (1975), 231–250.
[M]
J. Mack, Directed cover and paracompact spaces, Canad. J. Math. 19 (1967), 649–654.

COVERING PROPERTIES OF INVERSE LIMITS, II

7(K. Chiba) Department of Mathematics, Faculty of Science, Shizuoka University,
Shizuoka, 422-8529, Japan
E-mail address: smktiba@ipc.shizuoka.ac.jp(Y. Yajima) Department of Mathematics, Kanagawa University, Yokohama 2218686, Japan
E-mail address: yuki@cc.kanagawa-u.ac.jp

Proceedings of General Topology Symposium, p. 9 - p. 12, 2002

http://gt2002.h.kobe-u.ac.jp/

TOPOLOGICAL SEQUENCE ENTROPY OF MONOTONE MAPS
ON ONE-DIMENSIONAL CONTINUANAOTSUGU CHINEN
Abstract. Let X be either a dendrite or a graph and f a monotone map fromX to itself. The main result is that every topological sequence entropy of f
respect to every sequence S is zero. This implies that the topological entropyof f is equal to zero.

1. Introduction.
Let f be a continuous map from a continuum X to itself. We denote the n-foldcomposition f n of f with itself by f ◦ · · · ◦ f and f 0 the identity map. Let us denotef −i (Y ) the ith inverse image of an arbitrary set Y ⊂ X.
T. N. T. Goodman introduced in [G] the notion of topological sequence entropy
as an extension of the concept to topological entropy. Let f be a continuous map
from a compact metric space X to itself and A, B finite open covers of X. Denote{f −m (A)|A ∈ A} by f −m (A) for each positive integer m, A ∨ B = {A ∩ B|A ∈A, B ∈ B} and N (A) denotes the minimal possible cardinality of a subcover chosenfrom A. Let S = {si |i = 1, 2, . . . } be an increasing unbounded sequence of positive
integers. We define the topological sequence entropy of f relative to a finite open
cover A of X (respect to the sequence S) ashS (f, A) = lim sup
n→∞

n−1
_
1
log N (
f −si (A)).
ni=1

And we define the topological sequence entropy of f (respect to the sequence S) ashS (f ) = sup{hS (f, A)|A is a finite open cover of X}.
If si = i for each i, then hS (f ) is equal to the standard topological entropy h(f )of f introduced by Adler, Konheim and McAndrew in [AKM]. And set
h∞ (f ) = sup hS (f ).
S

If X is a compact interval or the circle, in [FS] and [H] it was proved that f ischaotic in the sense of Li and Yoke if and only if h∞ (f ) > 0. If f : [0, 1] → [0, 1]
is piecewise monotonic, Cánovas proved in [C1] that f is chaotic of type 2∞ if andonly if h∞ (f ) = log 2, and that f is chaotic of type greater than 2∞ if and only if
h∞ (f ) = ∞.
2000 Mathematics Subject Classification. Primary 37B40, 37E10; Secondary 28D05, 54H20.
Key words and phrases. Topological sequence entropy ; regular continuum ; dendrite ; graph.
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NAOTSUGU CHINEN

The mesh of an open cover A of X is the supremum of the diameter of theelementsof A, denoted by meshA. For an open cover A of X, we set Bd(A) =S{Bd(A)|A ∈ A}, where Bd(Y ) denotes the boundary of Y in X. A continuum X
is said to be regular if for each ε > 0, there exists a finite open cover A of X with
meshA < ε such that Bd(A) is finite. A locally connected continuum is said to bea dendrite if it contains no simple closed curve. See [N] for dendrites. It is known
that all dendrites and all graphs are regular.
Seidler proved in [S] the following theorem.
Theorem 1.1. If f is a homeomorphism from a regular continuum X into itself,then the topological entropy h(f ) of f is equal to zero.
This implies that if X is either a dendrite or a graph, then h(f ) = 0 for all
homeomorphism f : X → X.
A continuous map f : X → X is said to be monotone if f −1 (Y ) is connected
for each connected subset Y of f (X). It is well known that f n is monotone foreach positive integer n if f is monotone. It follows from [KS, Theorem D] that if
X is either a compact interval or the circle, then h∞ (f ) = 0 for all monotone map
f : X → X. And Efremova and Markhrova in [EM] considered some class D ofdendrites and showed that the topological entropy h(f ) of f is equal to zero for all
monotone map f from X ∈ D to itself. It is known from [BC] that h(f ) = h(f |Ω(f ) )for all continuous map f from a compact space to itself, where Ω(f ) denotes the
non-wandering set of f . In fact, it is proved in [EM] that h(f |Ω(f ) ) = 0 for all
monotone map f from X ∈ D to itself. But by [C2], there exists a continuous map
f : [0, 1] → [0, 1] such that hS (f ) > hS (f |Ω(f ) ) for a suitable sequence S of positive
integers. This shows that the topological sequence entropy is different from the
topological entropy.
We show the following theorem which is an extension of [KS, Theorem D] and[EM, Theorem B(B4)].
Theorem 1.2. Let X be either a dendrite or a graph and f a monotone map from
X into itself. Then h∞ (f ) = 0. In particular, the topological entropy of f is equal
to zero.
2. Definitions.
Definition 2.1. Let Y be a subspace of a metric space X. Cl(Y ) and diamY
denote the closure and the diameter of Y in a space X, respectively.
The cardinality of a set P will be denoted by Card(P ).
Definition 2.2. We say that a cover B is finer than a cover A, andWwrite B ≥ A ifneach
∈ B is contained in some A ∈ A. Clearly, if B ≥ A, then i=1 f −si (B) ≥Wn B−si(A) for any finite sequence s1 , s2 , . . . , sn of positive integers, and N (B) ≥i=1 f
N (A).
Definition 2.3. Let f be a continuous map from a continuum X to itself
T∞and n apositive integer. Denote Df,n = {x ∈ X|Card(f −n (x)) ≤ 1} and Df = n=1 Df,n .
Since Df,n is a Gδ set for each n, we see that Df is so. A continuum X is said
to be regular for f if for each ε > 0 there exists a finite open cover A of X with
meshA < ε such that Bd(A) is finite contained in Df for each A ∈ A.

TOPOLOGICAL SEQUENCE ENTROPY OF MONOTONE MAPS

11

3. Elementary lemmas.
Lemma 3.1. Let f be a continuous map from a continuum X to itself, S = {si |i =1, 2, . . . } an increasing unbounded sequence of positive integers and {An } a sequenceof finite open covers of X with limn→∞ mesh(An ) = 0 and An+1 ≥ An for all n.
Then hS (f ) = limn→∞ hS (f, An ).
Corollary 3.2. Let f be a continuous map from a continuum X to itself, S ={si |i = 1, 2, . . . } an increasing unbounded sequence of positive integers. If for eachε > 0 there exists a finite open cover of X with meshA < ε and hS (f, A) = 0, thenhS (f ) = 0.
Lemma 3.3. Let f be a continuous map from a continuum X to itself, n a positive integer, A an open cover of X with Card(A) ≥ 2, s1 , s2 , . . . , sn−1 a seWn−1
quence of positive integers and B a subcover of i=1 f −si (A). Then Bd(B) ⊂Sn−1
Bd( i=1 f −si (A)).
Lemma 3.4. Let f be a continuous map from a continuum X to itself and A a
finite open cover of X with Card(A) ≥ 2 such that Bd(A) is finite contained inP
Wn−1
Df . And let LA = A∈A Card(Bd(A)). Then N ( i=1 f −si (A)) ≤ nLA for any
sequence s1 , s2 , . . . , sn−1 of positive integers.
Lemma 3.5. Let f be a monotone map from a regular continuum X to itself. Then
Df is a Gδ dense set in X.
4. A proof of Theorem 1.2.
Theorem 4.1. Let f be a monotone map from X into itself. If X is regular for
f , then h∞ (f ) = 0. In particularly, h(f ) = 0.
Corollary 4.2. Let f be a continuous map from a regular continuum X to itself.
If f is embedding, then h∞ (f ) = 0.
Proof of Theorem 1.2. Let X be a dendrite or a graph and f a monotone map
from X to itself. Lemma 3.5 implies Df is a Gδ dense set. Let ε > 0. Since X is adendrite or a graph, there exists a finite set F ⊂ Df such that diamBS < 1/2ε for all
B ∈ B, where B is the set of all components of X \ F . Set Ax = {{x} ∪ B|B ∈ B
and x ∈ Cl(B)} for each x ∈ F and A = {Ax |x ∈ F }. We see that A is a finite
open cover of X with meshA < ε such that Bd(A) ⊂ F ⊂ Df . This shows that X
is regular for f . It follows from Theorem 4.1 that h∞ (f ) = 0.
¤
From the proof of Theorem 1.2, we have the following corollary.
Corollary 4.3. Let f be a continuous map from a regular continuum X to itself.
If for each ε > 0 there exists a finite subset F of Df such that diamB < ε for allcomponent B of X \ F , then h∞ (f ) = 0.
References
[AKM]
[BC]
[C1]

R.L. Adler, A.G. Knnhein and M.H. McAndrew, Topological entropy, Trans. Amer.
Math. Soc., 114 (1965), 309–319.
L. Block and W. Coppel, Dynamics in One Dimension, Lecture Notes in Math. 1513,
Springer-Verlag, 1992.
J. Cánovas, On topological sequence entropy of piecewise monotonic mappings, Bull.
Austral. Math. Soc., 62 (2000), 21–28.

12

NAOTSUGU CHINEN

[C2]
[EM]
[FS]
[G]
[H]
[KS]
[N]
[S]

J. Cánovas, An interval counterexample on topological sequence entropy, Acta Math.
Hungar., 88 (2000), 123–131.
L. S. Efremova and E. N. Makhrova, The dynamics of monotone maps of dendrites, Sb.
Math., 192 (2001), 807–821.
N. Franzová and J. Smı́tal, Positive sequence topological entropy characterizes chaotic
maps, Proc. Amer. Math. Soc., 112 (1991), 1083–1086.
T. N. T. Goodman, Topological sequence entropy, Proc. London Math. Soc., 29 (1974),
331–350.
R. Hric, Topological sequence entropy for maps of the circle, Comment. Math. Univ.
Carolinae, 41 (2000), 53–59.
S. Kolyada and L’. Snoha, Topological entropy of nonautonomous dynamical systems,
Random and Comp. Dynamics, 4 (1996), 205–233.
S.B. Nadler Jr, Continuum Theory An Introduction, Pure and Appl.Math.158(1992).
G. T. Seidler, The topological entropy of homeomorphisms on one-dimensional continua,
Proc. Amer. Math. Soc., 108 (1990), 1025–1030.

Institute of Mathematics, University of Tsukuba, Ibraki 305-8571 Japan
E-mail address: naochin@math.tsukuba.ac.jp

Proceedings of General Topology Symposium, p. 13 - p. 27, 2002

http://gt2002.h.kobe-u.ac.jp/

HEREDITARILY SEPARABLE GROUP TOPOLOGIES ON
ABELIAN GROUPSDIKRAN DIKRANJAN AND DMITRI SHAKHMATOV

Dedicated to the memory of Zoltán T. Balogh
All group topologies in this paper are considered to be Hausdorff (and thus
Tychonoff). Recall that a topological space X is:
Lindelöf if every open cover of X has a countable subcover,(countably) compact if every (countable) open cover of X has a finite subcover,pseudocompact if every real-valued continuous function defined on X is bounded,andseparable if X has a countable dense subset.
It is well-known that compact → countably compact → pseudocompact, and“pseudocompact + Lindelöf” ↔ compact.
Recall that a topological group G is precompact, or totally bounded , if G is
(topologically and algebraically isomorphic to) a subgroup of some compact group.
Pseudocompact groups are precompact [7], so we have a somewhat longer chain
compact → countably compact → pseudocompact → precompact
of compactness-like conditions for topological groups.
A space X is called hereditarily separable if every subspace of X is separable (inthe subspace topology), and X is said to be hereditarily Lindelöf if every subspaceof X is Lindelöf (in the subspace topology). An S-space is a hereditarily separable
regular space that is not Lindelöf.
1. Motivation
Our results originate in three diverse areas of mathematics.
The first source of inspiration comes from the celebrated theory of S-spaces
in set-theoretic topology, and especially, a famous 1975 example of Fedorčuk ofa hereditarily separable compact space of size 2c . In our paper we completely
characterize Abelian groups that admit a group topology making them into an Sspace, and we produce the “best possible analogues” of the Fedorčuk space in the1991 Mathematics Subject Classification. Primary: 22A05; Secondary: 03E35, 03E99, 54A20,
54A35, 54D05, 54D10, 54D20, 54D30, 54D65, 54G15, 54G99, 54H11.
Key words and phrases. topological group, countably compact, pseudocompact, hereditarily
separable, convergent sequence, S-space, consistency results, independence results, forcing.
Presented at the conference by the second author.
This project was started while the first named author was visiting the Department of Mathematics of Ehime University at Matsuyama in July 2002. He takes the opportunity to thank his
hosts for the generous hospitality and support.
13

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D. DIKRANJAN AND D. SHAKHMATOV

category of topological groups. As it turns out, a vast majority of Abelian groups
admit group topologies with properties similar to that of the Fedorčuk example.
The second origin lies in topological algebra, where we were motivated by the
problem of which Abelian groups admit a countably compact group topology. We
completely describe, albeit consistently, the algebraic structure of Abelian groupsof size at most 2c that admit a countably compact group topology.
Our third motivation comes from the theory of cardinal invariants in general
topology. We resolve completely a 1980 problem of van Douwen about the cofinality
of |G| for a countably compact group G in the case of Abelian groups.
We will now address all three sources of our motivation in detail.
1.1. S-groups à la Fedorčuk. Recall that |Y | ≤ c for a hereditarily Lindelöf
Hausdorff space Y [1], and |X| ≤ 2c for a separable Hausdorff space X [32]. It
is natural to ask whether the last inequality can be strengthened to |X| ≤ c fora hereditarily separable regular space X. If there are no S-spaces, then every
hereditarily separable regular space X is hereditarily Lindelöf, and therefore |X| ≤ cby the result cited above. Todorčević has proved the consistency with ZFC that
S-spaces do not exist ([40], see also [41]). Therefore, in Todorčević’s model of ZFC,
hereditarily separable regular spaces have size at most c. A first consistent example
of a hereditarily separable Tychonoff space of size 2c has been found by Hajnal andJuhász [20]. Two years later Fedorčuk [17] produced the strongest known example
up to date using his celebrated inverse spectra with fully closed maps (see also [18]):
Example 1.1. The existence of the following “Fedorčuk space” X is consistent
with ZFC plus CH:(i) |X| = 2c ,(ii) X is hereditarily separable,(iii) X is compact, and(iv) if F is an infinite closed subset of X, then |F | = |X|; in particular, X does
not contain non-trivial convergent sequences.
The main goal of this paper is to address the question of the existence of “Fedorčuk space” in the context of topological groups. That is, given a group G, we
wonder if it is possible to find a hereditarily separable Hausdorff group topology on
G having properties that “Fedorčuk space” has. Since we want to get a hereditarily
separable topology on G, we have to restrict ourselves to groups G of size at most
2c . One naturally expects that the presence of algebra may produce additional
restrictions on how good a Fedorčuk type group can be. And this is indeed thecase.
First of all, one is forced to relax somewhat the compactness condition from item(iii) of Example 1.1 because of two fundamental facts about compact groups:
Fact 1.2. (i) Infinite compact groups contain non-trivial convergent sequences.
(ii) Compact hereditarily separable groups are metrizable.
Both facts are folklore and follow from the following result of Hagler, Gerlits andEfimov: An infinite compact group G contains a copy of the Cantor cube {0, 1}w(G) ,where w(G) is the weight of G. An elementary proof of this theorem, together withsome historical discussion, can be found in [35].
Recall that a space X is initially ω1 -compact if every open cover of size ≤ ω1 hasa finite subcover. Item (i) of Fact 1.2 is no longer valid, at least consistently, if one

HEREDITARILY SEPARABLE GROUP TOPOLOGIES

15

replaces “compact” by “initially ω1 -compact” in it: It is consistent with ZFC that
there exists an initially ω1 -compact Hausdorff group topology without non-trivial
convergent sequences on the free Abelian group of size c. This result is announced,
with a hint at a proof, in [42].
However, item (ii) of Fact 1.2 remains valid if one replaces “compact” by “initiallyω1 -compact” in it, see [2]. This means that countable compactness appears to be thestrongest compactness type property among weakenings of classical compactness forwhich one may hope to obtain hereditarily separable group topologies, and indeed,
consistent examples of hereditarily separable countably compact groups (without
non-trivial convergent sequences) are known in the literature [21, 38, 28]. This
perfectly justifies countable compactness as our strongest compactness condition of
choice when working with hereditarily separable groups.
Second, we will have to restrict ourselves to Abelian groups because in the noncommutative case there are groups (of small size) that do not admit any countablycompact or separable group topology, as follows from our next result:
Proposition 1.3. Let X be a set and S(X) the symmetric group of X.1 Then:(i) S(X) does not admit a separable group topology unless X is countable,(ii) S(X) admits no countably compact group topology when X is infinite, and(iii) S(X) does not admit a Lindelöf group topology unless X is countable.
Proof. We equip S(X) with the topology of pointwise convergence on X, i.e. thetopology Tp generated by the family {U (f, F ) : f ∈ S(X), F ∈ [X]<ω } as a base,where U (f, F ) = {g ∈ S(X) : g(x) = f (x) for all x ∈ F }. It is easy to see that Tp
is a group topology.
Assume that X is an infinite set. For a fixed x ∈ X, the stabilizer Sx = {σ ∈S(X) : σ(x) = x} = U (idX , {x}) of x is a Tp -open subgroup of S(X) of index |X|,and hence it produces an open cover of S(X) by pairwise disjoint sets (obtained bytaking appropriate unions of cosets of Sx ) without a subcover of size (strictly) lessthan |X|. It follows that the space (S(X), Tp ) is not countably compact, and also
is neither separable nor Lindelöf when |X| > ω.
It is known that Tp is a minimal element in the lattice of all (Hausdorff) group
topologies on S(X), i.e. Tp ⊆ T for every (Hausdorff) group topology T on S(X)[19]. This easily yields the conclusion of all three items of our proposition.
¤
It follows from the above proposition that, for an uncountable set X, the symmetric group S(X) admits neither a separable, nor a countably compact, nor a Lindelöfgroup topology.2 Furthermore, free groups never admit countably compact group
topologies ([10, Theorem 4.7]; see also [12, Corollary 5.14]).
Third, algebraic restrictions prevent us from getting the full strength of item(iv), as our next example demonstrates:
1That is, S(X) is a set of bijections of X onto itself with the composition of maps as
multiplication.
2In particular, no group S(X) admits a Hausdorff group topology that makes it into an Sspace. This should be compared with substantial difficulties one has to overcome to produce amodel of ZFC in which there are no S-spaces. Furthermore, no group S(X) admits a Hausdorffgroup topology that makes it into an L-space (i.e., a hereditarily Lindelöf but not (hereditarily)
separable space). This should be compared with the fact that the consistency of the non-existenceof L-spaces is a well-known problem of set-theoretic topology that remains unresolved.

16

D. DIKRANJAN AND D. SHAKHMATOVc

Example 1.4. Let G = Z(2)(c) ⊕ Z(2 ) be the direct sum of the Boolean groupc
Z(2)(c) of size c and the free Abelian group Z(2 ) of size 2c . We claim that, for
any Hausdorff group topology on G, there exists a closed (in this topology) infinite
set F such that |F | < |G|. In fact, F = Z(2)(c) ⊆ G is such a set. Indeed,|F | = c < 2c = |G|, so it remains only to note that F is an unconditionally closed
subset of G in Markov’s sense [31]; that is, F is closed in every Hausdorff group
topology on G. The latter follows from the fact that F = {x ∈ G : 2x = 0} is thepreimage of the (closed!) set {0} under the continuous map that sends x to 2x.
We note that our Theorem 2.7 implies that, in an appropriate model of ZFC,
the group G from the example above does admit a hereditarily separable countablycompact group topology without non-trivial convergent sequences. So the best we
can hope for in our quest for Fedorčuk type group G is to require that G satisfies
the second, weaker, condition from item (iv) of Example 1.1, i.e. that G does nothave any non-trivial convergent sequences. In fact, we will manage to get a stronger
condition: G does not have infinite compact subsets.
1.2. Algebraic structure of countably compact Abelian groups. Halmos [22] showed that the additive group of real numbers can be equipped witha compact group topology and asked which Abelian groups admit compact group
topologies. Halmos’ problem seeking a complete description of the algebraic structure of compact Abelian groups contributed substantially to the development of
the Abelian group theory, particularly through the introduction of the algebraicallycompact groups by Kaplansky [26]. This problem has been completely solved in[23, 24].
The counterpart of Halmos’ problem for pseudocompact groups asking which
Abelian groups can be equipped with a pseudocompact group topology was attacked
in [3, 10, 11, 4, 5, 12] and the significant progress has been summarized in themonograph [12]. Recall also that every Abelian group admits a precompact group
topology [6].
The question of which Abelian groups admit a countably compact group topology
appears to be much more complicated. After a series of scattered results [21, 15, 38,
28, 13, 43] a complete description of the algebraic structure of countably compact
Abelian groups of size at most c under Martin’s Axiom MA has been recently
obtained in [14]: MA implies that an Abelian group G of size at most c admits
a countably compact group topology if and only if it satisfies both PS and CC,
two conditions introduced in Definition 2.3 below. (In particular, every torsion-free
Abelian group of size c admits a countably compact group topology under MA[39].) In our Theorem 2.7 and Corollary 2.17(ii) we substantially extend this result
by proving that, at least consistently, the conjunction of PS and CC is both anecessary and a sufficient condition for the existence of a countably compact group
topology on an Abelian group G of size at most 2c . Moreover, we get both hereditary
separability and absence of infinite compact subsets for our group topology as abonus.
This “jump” from c to 2c is an essential step forward. Indeed, amazingly little
is presently known about the existence of countably compact group topologies ongroups of cardinality greater than c. Using a standard closing-off argument van
Douwen [16] showed that every infinite Boolean group of size κ = κω admits acountably compact group topology and his argument can easily be extended toHEREDITARILY SEPARABLE GROUP TOPOLOGIES

17

Abelian groups of prime exponent. It is consistent with ZFC that the Booleangroup of size κ has a countably compact group topology provided that c ≤ κ ≤ 2c[44]. (Here 2c can be made “arbitrary large”.) It is also consistent with ZFC thatthe free Abelian group of size κ has a countably compact group topology provided
that c ≤ κ = κω ≤ 2c [27]. Finally, it is well-understood which Abelian groups
admit compact group topologies. Essentially these are the only known results inthe literature about the existence of countably compact group topologies on groupsof cardinality greater than c (even without the additional requirement of hereditary
separability).
While the algebraic description of Abelian groups admitting either a compact
or a pseudocompact group topology can be carried out without any additional
set-theoretic assumptions beyond ZFC, all known results about countably compact topologizations described above have either been obtained by means of some
additional set-theoretic axioms (usually Continuum Hypothesis CH or versions ofMartin’s Axiom MA) or their consistency has been proved by forcing. Even thefundamental question (raised in [38]) as to whether the free Abelian group of sizec admits a countably compact group topology is still open in ZFC. (Recall that nofree Abelian group admits a compact group topology.)
It seems worth noting a peculiar difference between compact and countably compact topologizations of Abelian groups. In the compact case the sufficiency of thealgebraic conditions is relatively easy to prove, whereas their necessity is muchharder to establish. In the countably compact case the necessity of PS and CC is
immediate (see Lemma 2.5), while the sufficiency is rather complicated and at thepresent stage requires additional set-theoretic assumptions.
1.3. van Douwen’s problem: Is |G| = |G|ω for a countably compact
group G? It is well-known that |G| = 2w(G) for an infinite compact group G,
where w(G) is the weight of G [25]. In particular, the cardinality |G| of an infinite
compact group G satisfies the equation |G| = |G|ω . This motivated van Douwen to
ask in [16] the following natural question: Does |G| = |G|ω , or at least cf(|G|) > ω,hold for every infinite topological group (or homogeneous space) G which is countably compact?
In the same paper [16] van Douwen proved that, under the Generalized Continuum Hypothesis GCH, every infinite pseudocompact homogeneous space G satisfies|G| = |G|ω . In particular, a strong positive answer (with countable compactness
weakened to pseudocompactness, and “topological group” weakened to “homogeneous space”) to van Douwen’s problem is consistent with ZFC. A first consistent
counter-example to van Douwen’s question was recently announced by Tomita [44]who used forcing to construct a model of ZFC in which every Boolean group of size
κ has a countably compact group topology provided that c ≤ κ ≤ 2c [44, Theorem2.2]. Here 2c can be made “arbitrary large” so that, for any given ordinal σ ≥ 1chosen in advance, one can arrange that c ≤ ℵσ ≤ 2c (in particular, ℵω can be
included in the interval between c and 2c ).
In our Corollary 2.23 we push Tomita’s negative solution to van Douwen’s question to the extreme limit by demonstrating that, in a sense, the cofinality of |G| fora countably compact Abelian group G is completely irrelevant: For every ordinalσ ≥ 1 it is consistent with ZFC that every Abelian group G of size ℵσ admits
a countably compact group topology provided that G satisfies PS and CC, two

18

D. DIKRANJAN AND D. SHAKHMATOV

necessary conditions for the existence of such a topology on G (see Definition 2.3
and Lemma 2.5(ii)).
2. Main results
The major achievement of this paper is a forcing construction of a (class of) special model(s) of ZFC in which Abelian groups of size at most 2c admit hereditarily
separable group topologies with various compactness-like properties and without
infinite compact subsets. Let us produce an outline of our construction.
We define, for every cardinal κ ≥ ω2 , a forcing notion (Pκ , ≤) that depends only
on this cardinal κ (thereby justifying the notation Pκ ). Let Mκ be an arbitrary
countable transitive model of ZFC satisfying κ ∈ Mκ , (Pκ , ≤) ∈ Mκ , c = ω1 and
2ω1 = κ. If G ⊆ Pκ is a Pκ -generic set over Mκ , then the generic extension Mκ [G]
has the same cardinals as Mκ and the equalities ω1 = c and 2ω1 = 2c = κ holdin Mκ [G]. Since the original cardinal κ can be taken to be “arbitrarily large”,
the power 2c of the continuum c in the generic extension Mκ [G] can also be made“arbitrarily large”.
In all results below, Mκ [G] will always denote the generic extension describedabove.
Our first main result shows that, at least consistently, the inequality |G| ≤ 2cis the only necessary condition for the existence of a hereditarily separable group
topology on an Abelian group:
Theorem 2.1. In Mκ [G], the following conditions are equivalent for any Abeliangroup G:(i) G admits a separable group topology,(ii) G admits a hereditarily separable group topology,(iii) G admits a hereditarily separable precompact group topology without infinite
compact subsets, and(iv) |G| ≤ 2c .
Recall that Todorčević constructed a model of ZFC in which S-spaces do notexist ([40], see also [41]). Things change dramatically in this model:
Theorem 2.2. In any model of ZFC in which there are no S-spaces the following
conditions are equivalent for any Abelian group G:(i) G admits a hereditarily separable group topology,(ii) G admits a separable metric precompact group topology, and(iii) |G| ≤ c.
We would like to emphasize that there are absolutely no algebraic restrictions(except natural restriction of commutativity) on the group G in the above two theorems. Algebraic constraints become more prominent when one adds some compactness condition to the mix.
Let G be an Abelian group. As usual r(G) denotes the free rank of G. For every
natural number n ≥ 1 define G[n] = {g ∈ G : ng = 0} and nG = {ng : g ∈ G}.
Recall that G is:S
torsion provided that G = {G[n] : n ∈ ω \ {0}},bounded torsion if G = G[n] for some n ∈ ω \ {0},torsion-free if G[n] = {0} for every n ∈ ω \ {0}, anddivisible if mG = G for each m ∈ ω \ {0}.

HEREDITARILY SEPARABLE GROUP TOPOLOGIES

19

We will now introduce three algebraic conditions that will play a prominent role
throughout this paper.
Definition 2.3. For an Abelian group G, define the following three conditions:
PS: Either r(G) ≥ c or G is a bounded torsion group.
CC: For every pair of integers n ≥ 1 and m ≥ 1 the group mG[n] is either finite
or has size at least c.
tCC: If G is torsion, then CC holds.
Our next lemma, despite its simplicity, is quite helpful for better understandingof these conditions:Lemma 2.4. Let G be an Abelian group.
(i) If G is torsion, then G satisfies PS if and only if G is a bounded torsion
group.
(ii) If G is a torsion-free group, then G satisfies PS if and only if |G| ≥ c.
(iii) If G is a torsion-free group, then G satisfies CC.
(iv) CC for G implies tCC.
(v) If G is not torsion, then G satisfies tCC.
(vi) If G is torsion and satisfies tCC, then G satisfies CC as well.
Proof. To prove (i) note that r(G) = 0 < c if G is torsion.
(ii) If G is a torsion-free group, then condition PS for G becomes equivalent tor(G) ≥ c, and the latter condition is known to be equivalent to |G| ≥ c.
(iii) Assume that G is torsion-free. Let n ≥ 1 and m ≥ 1 be natural numbers.
Then G[n] = {0} and hence mG[n] = {0} is finite. Therefore CC holds.
Items (iv), (v) and (vi) are trivial.
¤
Condition PS is known to be necessary for the existence of a pseudocompact
group topology on an Abelian group G, thereby justifying its name (PS stands
for “pseudocompact”). To the best of the author’s knowledge, this fact has beenannounced without proof in [3, Remark 2.17] and [10, Proposition 3.3], and has
appeared in print with full proof in [12, Theorem 3.8].
It can be easily seen that condition CC is necessary for the existence of a countably compact group topology on an Abelian group G, thereby justifying its name(CC stands for “countably compact”). Indeed, if G is a countably compact group,
then the set G[n] = {g ∈ G : ng = 0} must be closed in G, and thus G[n] is countably compact in the subspace topology induced on G[n] from G. Furthermore, themap which sends g ∈ G[n] to mg ∈ mG[n] is continuous, and so mG[n] must be
countably compact (in the subspace topology). It remains only to note that an
infinite countably compact group has size at least c [16, Proposition 1.3 (a)]. In theparticular case when an Abelian group G has size c, the fact that CC is a necessary
condition for the existence of a countably compact group topology on G has been
proved in [14].
Condition CC has essentially appeared for the first time in [10] where it wasproved that CC is necessary for the existence of a pseudocompact group topology
on a torsion Abelian group.3 Since CC and tCC are equivalent for torsion groups
by items (v) and (vii) of Lemma 2.4, it follows that tCC is a necessary condition
3Furthermore, it is proved in [10] that CC is also a sufficient condition for the existence of apseudocompact group topology on a bounded torsion Abelian group of size at most 2c . See also
the proof of Theorem 2.22.

20

D. DIKRANJAN AND D. SHAKHMATOV

for the existence of a pseudocompact group topology on a torsion group, thereby
justifying our choice of terminology (tCC stands for “torsion CC”). Since tCC
trivially holds for non-torsion groups (see item (vi) of Lemma 2.4), we conclude
that tCC is a necessary condition for the existence of a pseudocompact group
topology on an Abelian group G.
We can now summarize the discussion above in a convenient lemma:
Lemma 2.5. (i) A pseudocompact Abelian group G satisfies PS and tCC.
(ii) A countably compact Abelian group G satisfies PS and CC.
In the “opposite direction”, it is known that the combination of PS and tCC
is sufficient for the existence of a pseudocompact group topology on an Abeliangroup G of size at most 2c ([10]; see also [12]) and, under Martin’s Axiom MA, thecombination of PS and CC is sufficient for the existence of a countably compact
group topology on an Abelian group G of size at most c [14].
In our next “twin” theorems we establish that these pairs of conditions are, consistently, also sufficient for the existence of a hereditarily separable pseudocompact
and countably compact group topology on a group G of size at most 2c .
Theorem 2.6. In Mκ [G], the following conditions are equivalent for any Abeliangroup G:(i) G admits a separable pseudocompact group topology,(ii) G admits a hereditarily separable pseudocompact group topology,(iii) G admits a hereditarily separable pseudocompact group topology without infinite compact subsets, and(iv) |G| ≤ 2c and G satisfies both PS and tCC.
We can also prove that the equivalence of items (i) and (iv) in the above theorem
holds in ZFC.
Theorem 2.7. In Mκ [G], the following conditions are equivalent for any Abeliangroup G:(i) G admits a separable countably compact group topology,(ii) G admits a hereditarily separable countably compact group topology,(iii) G admits a hereditarily separable countably compact group topology without
infinite compact subsets, and(iv) |G| ≤ 2c and G satisfies both PS and CC.
Theorem 2.7 recovers (and greatly extends) the main result of Dikranjan andTkachenko [14]: It is consistent with ZFC that an Abelian group G of size at mostc has a countably compact group topology if and only if G satisfies both PS andCC.
Things become “essentially trivial” in Todorčević’s model of ZFC without Sspaces:
Theorem 2.8. In any model of ZFC in which there are no S-spaces the following
conditions are equivalent for any Abelian group G:(i) G admits a hereditarily separable pseudocompact group topology,(ii) G admits a hereditarily separable countably compact group topology, and(iii) G admits a compact metric group topology.
Let G be any Abelian group such that c < |G| ≤ 2c . Since compact metricspaces have size at most c, our previous theorem implies that, consistently, G does

HEREDITARILY SEPARABLE GROUP TOPOLOGIES

21

not admit a hereditarily separable pseudocompact group topology. On the other
hand, if one additionally assumes that G satisfies both PS and CC, then G admitsa hereditarily separable countably compact group topology in the model Mκ [G](Theorem 2.7). In particular, we conclude that the existence of a hereditarily separable pseudocompact (or countably compact) group topology on the free Abeliangroup of size 2c is both consistent with and independent of ZFC . (An example of
an Abelian group of size c with similar properties is much harder to obtain.)
We will now look at what our Theorems 2.6 and 2.7 say for four particular important subclasses of Abelian groups: torsion groups, non-torsion groups, torsion-freegroups, and divisible groups.
Corollary 2.9. In Mκ [G], the following conditions are equivalent for any torsion
Abelian group G:(i) G admits a separable pseudocompact group topology,(ii) G admits a hereditarily separable countably compact group topology without
infinite compact subsets, and(iii) |G| ≤ 2c and G is a bounded torsion group satisfying CC.
Proof. Let G be a torsion Abelian group. According to Lemma 2.4(i), a bounded
torsion group satisfies PS, so (iii) implies (ii) by Theorem 2.7. The implication (ii)
→ (i) is trivial. To see that (i) → (iii), note that |G| ≤ 2c and G satisfies both
PS and tCC by Lemma 2.5(i). Since G is torsion, Lemma 2.4(i) yields that G isa bounded torsion group, while Lemma 2.4(vi) implies that G satisfies CC.
¤
The following particular case of the above corollary seems to be worth mentioning:
Corollary 2.10. In Mκ [G], for every prime number p, each natural number n ≥ 1
and every infinite cardinal τ , the following conditions are equivalent:(i) Z(pn )(τ ) admits a separable pseudocompact group topology,(ii) Z(pn )(τ ) admits a hereditarily separable countably compact group topology
without infinite compact subsets, and(iii) c ≤ τ ≤ 2c .
Proof. For the group Z(pn )(τ ) , condition CC is equivalent to “τ is either finite orτ ≥ c”, and the result follows from Corollary 2.9.
¤
Since torsion pseudocompact groups are always zero-dimensional [9], the assumption that G is non-torsion is necessary in the next two theorems.
Theorem 2.11. In Mκ [G], the following conditions are equivalent for any nontorsion Abelian group G:(i) G admits a separable pseudocompact group topology,(ii) G admits a hereditarily separable connected and locally connected pseudocompact group topology without infinite compact subsets, and(iii) |G| ≤ 2c and G satisfies PS.
Theorem 2.12. In Mκ [G], the following conditions are equivalent for any nontorsion Abelian group G:(i) G admits a separable countably compact group topology,(ii) G admits a hereditarily separable connected and locally connected countablycompact group topology without infinite compact subsets, and(iii) |G| ≤ 2c and G satisfies PS and CC.

22

D. DIKRANJAN AND D. SHAKHMATOV

In the case of torsion-free groups things become very transparent, as algebraic
restraints disappear again:
Corollary 2.13. In Mκ [G], the following conditions are equivalent for any torsionfree Abelian group G:(i) G admits a separable pseudocompact group topology,(ii) G admits a hereditarily separable countably compact connected and locallyconnected group topology without infinite compact subsets, and(iii) c ≤ |G| ≤ 2c .
Proof. Let G be a torsion-free Abelian group. According to item (iii) of Lemma
2.4, condition PS for G is equivalent to |G| ≥ c, while items (iv) and (v) of thesame lemma imply that both conditions CC and tCC hold for G. It remains only
to plug these facts into Theorems 2.11 and 2.12.
¤
We note that even a very particular case of our Corollary 2.13 constitutes themain result of Koszmider, Tomita and Watson [27]: It is consistent with ZFC
that for every cardinal τ such that c ≤ τ = τ ω ≤ 2c the free Abelian group of
size τ admits a countably compact group topology without non-trivial convergent
sequences. The topology constructed in [27] is not hereditarily separable, while
our topology is. Furthermore, while our topology does not have infinite compact
subsets, it is not at all clear if the topology from [27] has infinite compact subsets
or not.
As usual, for a prime number p and an Abelian group G, rp (G) denotes the p-rankof G. Our next theorem reduces the problem of the existence of a (hereditarily)
separable countably compact group topology on a divisible Abelian group G to asimple checking of transparent conditions involving the cardinality, free rank andp-ranks of G.
Theorem 2.14. In Mκ [G], the following conditions are equivalent for any nontrivial divisible Abelian group G:(i) G admits a separable countably compact group topology,(ii) G admits a hereditarily separable connected and locally connected countablycompact group topology without infinite compact subsets,(iii) c ≤ r(G) ≤ |G| ≤ 2c and, for every prime number p, either the p-rank rp (G)of G is finite or the inequality rp (G) ≥ c holds.
Corollary 2.15. In Mκ [G], the following conditions are equivalent for any Abeliangroup G:(i) G admits a separable connected precompact group topology,(ii) G admits a hereditarily separable connected and locally connected pseudocompact group topology without infinite compact subsets.
Proof. (i) →(ii). Since G is precompact, there exists a non-trivial continuous character χ : G → T. Then χ(G) is a non-trivial connected subgroup of T, which yields
χ(G) = T. Therefore r(G) ≥ r(T) = c. In particular, G is non-torsion and satisfies
PS. The separability of G yields |G| ≤ 2c . Now implication (iii)→(ii) of Theorem 2.11 guarantees that G admits a hereditarily separable connected and locallyconnected pseudocompact group topology without infinite compact subsets.
(i) →(ii) is trivial.
¤
Fact 1.2(i) inspired a quest for constructing compact-like group topologies without non-trivial convergent sequences, see, for example, [36, 21, 15, 28, 30, 38, 8, 44].

HEREDITARILY SEPARABLE GROUP TOPOLOGIES23

Our next corollary shows that, in a certain sense, one does not need to work that
hard in order to get these topologies: Indeed, at least on Abelian groups of size at
most 2c , there are “plenty” of them around in the model Mκ [G]:Corollary 2.16. In Mκ [G], let G be an Abelian group of size at most 2c . Then:(i) G admits a hereditarily separable precompact group topology without infinite
compact subsets,(ii) if G admits a pseudocompact group topology, then G also has a hereditarily
separable pseudocompact group topology without infinite compact subsets,(iii) if G admits a countably compact group topology, then G also has a hereditarily separable countably compact group topology without infinite compact subsets.
Proof. Item (i) follows from the implication (iv) → (iii) of Theorem 2.1. Item (ii)follows from Lemma 2.5(i) and the implication (iv) → (iii) of Theorem 2.6. Item(iii) follows from Lemma 2.5(ii) and the implication (iv) → (iii) of Theorem 2.7. ¤As a by-product of our results, we can completely describe the algebraic structure of the Abelian groups of size at most 2c which admit, at least consistently, acountably compact group topology.
Corollary 2.17. In Mκ [G], let G be an Abelian group of size at most 2c . Then G
admits a countably compact group topology if and only if G satisfies both PS andCC.
Proof. The “only if” part follows from Lemma 2.5(ii), and the “if” part follows
from the implication (iv) → (iii) of Theorem 2.7.
¤
Corollary 2.18. In Mκ [G], a torsion Abelian group G of size at most 2c admits acountably compact group topology if and only if G is bounded and satisfies CC.
Proof. The “only if” part follows from Lemma 2.5(ii), and the “if” part follows
from the implication (iii) → (ii) of Corollary 2.9.
¤
Corollary 2.19. In Mκ [G], a torsion-free Abelian group G of size at most 2c admits
a countably compact group topology if and only if |G| ≥ c.
Proof. Corollary 2.13 applies.

¤

Corollary 2.20. In Mκ [G], the following two conditions are equivalent for every
Abelian group G of size at most 2c that is either torsion or torsion-free:(i) G admits a pseudocompact group topology, and(ii) G admits a countably compact group topology.
Proof. Clearly (ii) implies (i). To prove the converse, assume (i). Then G satisfies
PS and tCC by Lemma 2.5(i). If G is torsion, G satisfies CC by item (vii) of
Lemma 2.4. If G is torsion-free, then G satisfies CC by item (iv) of Lemma 2.5.
Since |G| ≤ 2c and G satisfies both PS and CC, Theorem 2.7 now yields that G
has a countably compact group topology.
¤
Corollary 2.21. In Mκ [G], a divisible Abelian group G of size at most 2c admits
a countably compact group topology if and only if r(G) ≥ c and, for every prime
number p, either the p-rank rp (G) of G is finite or the inequality rp (G) ≥ c holds.
Proof. This immediately follows from Theorem 2.14.

¤

24

D. DIKRANJAN AND D. SHAKHMATOV

The counterpart of Corollary 2.17 for pseudocompact group topologies can beproved in ZFC.
Theorem 2.22. Let G be an Abelian group of size at most 2c . Then G admits a
pseudocompact group topology if and only if G satisfies both PS and tCC.
We will now exhibit an application of Theorem 2.7 to van Douwen’s problem, see
Subsection 1.3. Our next corollary demonstrates that, contrary to van Douwen’s
belief, it is consistent with ZFC that there is nothing exceptional about Abeliangroups whose size has countable cofinality, such as ℵω , ℵω+ω , ℵω+ω+ω etc., fromthe point of view of the existence of countably compact group topologies.
Corollary 2.23. For every ordinal σ ≥ 1, it is consistent with ZFC and c = ω1that every Abelian group of size ℵσ satisfying conditions PS and CC admits a(hereditarily separable) countably compact group topology (without infinite compact
subsets).
Proof. “Make” κ bigger than ℵσ . Then, in Mκ [G], 2c = κ will also be bigger thanℵσ . Now our corollary immediately follows from the conclusion of Theorem 2.7. ¤Again, things become especially transparent in both torsion and torsion-freecase.
Corollary 2.24. For every ordinal σ ≥ 1, it is consistent with ZFC plus c = ω1 thatevery bounded torsion Abelian group of size ℵσ satisfying CC admits a (hereditarily
separable) countably compact group topology (without infinite compact subsets).
Proof. This follows from Corollary 2.23 because bounded torsion groups satisfy PS(see item (ii) of Lemma 2.4).
¤
Corollary 2.25. For every ordinal σ ≥ 1, it is consistent with ZFC plus c = ω1that for every prime number p and each natural number n ≥ 1 the group Z(pn )(ℵσ )admits a (hereditarily separable) countably compact group topology (without infinite
compact subsets).
Proof. This follows from Corollary 2.24 since the group Z(pn )(ℵσ ) satisfies condition
CC because c = ℵ1 ≤ ℵσ .
¤Even a particular case of our last corollary, with p = 2 and n = 1, implies themain result of Tomita [44]: For every ordinal σ ≥ 1, it is consistent with ZFC
plus c = ω1 that the Boolean group Z(2)(ℵσ ) of size ℵσ can be equipped with acountably compact group topology. It is also worth mentioning that the group
topology constructed in [44] is not hereditarily separable and has non-trivial convergent sequences (because it contains a Σ-product of uncountably many compact
metric groups, and it is easily seen that such a Σ-product is not separable and has
an infinite compact metric subgroup).
Corollary 2.26. For every ordinal σ ≥ 1, it is consistent with ZFC that every
torsion-free Abelian group of size ℵσ admits a (hereditarily separable) countablycompact group topology (without infinite compact subsets).
Proof. “Make” κ bigger than ℵσ . Then, in Mκ [G], 2c = κ will also be bigger thanℵσ . Since c = ℵ1 ≤ ℵσ , Corollary 2.13 applies.
¤HEREDITARILY SEPARABLE GROUP TOPOLOGIES25

Our results on hereditary separable topologizations allow us to make a contribution to the celebrated “S-space problem” . Scattered examples of topologicalgroups which are S-spaces are known in the literature [21, 15, 38, 28, 33, 34, 29, 37].
Our final three theorems describe completely which Abelian groups admit group
topologies (with various compactness conditions) which make them into S-spaces.
Theorem 2.27. In Mκ [G], the following are equivalent for an Abelian group G:(i) G admits a group topology that makes it into an S-space,(ii) G admits a precompact group topology that makes it into an S-space,(iii) c ≤ |G| ≤ 2c .
Theorem 2.28. In Mκ [G], the following are equivalent for an Abelian group G:(i) G admits a pseudocompact group topology that makes it into an S-space,(ii) c ≤ |G| ≤ 2c and G satisfies both PS and tCC.
Theorem 2.29. In Mκ [G], the following are equivalent for an Abelian group G:(i) G admits a countably compact group topology that makes it into an S-space,(ii) c ≤ |G| ≤ 2c and G satisfies both PS and CC.
Since hereditarily separable (initially ω1 -)compact groups are metrizable, a (initially ω1 -)compact group cannot be an S-space.
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Wetensch. Amsterdam 14 (1929).
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[10] D. Dikranjan and D. Shakhmatov, Pseudocompact topologizations of groups, Zb. radova Filozof. fakulteta u Nišu, Ser. Mat. 4 (1990), 83–93.
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[13] D. Dikranjan and M. Tkachenko, Varieties generated by countably compact abelian groups,
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[15] E. K. van Douwen, The product of two countably compact topological groups, Trans. Amer.
Math. Soc. 262 (1980), 417–427.
[16] E. K. van Douwen, The weight of a pseudocompact (homogeneous) space whose cardinality
has countable cofinality, Proc. Amer. Math. Soc. 80 (1980), 678–682.

26

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[22] P. Halmos, Comments on the real line, Bull. Amer. Math. Soc. 50 (1944), 877–878.
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366–391.
[24] A. Hulanicki, Algebraic structure of compact abelian groups, Bull. Acad. Polon. Sci. Sér. Sci.
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Tokyo 19 (1943), 366–372.
[26] I. Kaplansky, Infinite abelian groups, University of Michigan Press, Ann Arbor, 1954. v+91
pp.
[27] P. B. Koszmider, A. H. Tomita and S. Watson, Forcing countably compact group topologies
on a larger free Abelian group, Topology Proc. 25 (Summer) (2000), 563–574.
[28] V. I. Malykhin, An example of a topological group, in: Topological spaces and their mappings(Latv. Gos. Univ., Riga, 1981), pp. 120–123 (in Russian, with English summary).
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[32] B. Pospı́šil, Sur la puissance d’un espace contenant une partie dense de puissance donnée,Časopis Pro Pěstovánı́ Matematiky a Fysiky 67 (1937), 89–96.
[33] J. Roitman, Easy S and L groups, Proc. Amer. Math. Soc. 78 (1980), no. 3, 424–428.
[34] D. B. Shakhmatov, S and L fields, Vestnik Moskov. Univ. Ser. I Matem. Mekh. (1986), no.
5, 72-73 (in Russian); English transl. in: Moscow Univ. Math. Bull. 41, no. 5 (1986), 66-68.
[35] D. B. Shakhmatov, A direct proof that every infinite compact group G contains {0, 1}w(G) ,Ann. New York Acad. Sci. 728 (1994), 276–283.
[36] S. M. Sirota, A product of topological groups and extremal disconnectedness, Mat. Sb. (N.S.)
79(121) (1969), 179–192.
[37] L. Soukup, Indestructible properties of S- and L-spaces, Topology Appl. 112 (2001), 245–257.
[38] M. G. Tkachenko, On countably compact and pseudocompact topologies on free Abeliangroups, Soviet Math. (Izv. VUZ) 34 (5) (1990), 79–86. Russian original in: Izv. Vyssh.
Uchebn. Zaved. Ser. Matem. 5 (1990), 68–75.
[39] M. Tkachenko and I. Yaschenko, Independent group topologies on Abelian groups, Topology
Appl. 122 (2002), 425–451.
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703–720.
[41] S. Todorčević, Partition problems in topology (Contemporary Mathematics, 84), Amer.
Math. Soc., Providence, RI, 1989.
[42] A. H. Tomita, The existence of initially ω1 -compact group topologies on free abelian groups
is independent of ZFC , Comment. Math. Univ. Carolin. 39 (1998), 401–413.
[43] A. H. Tomita, On the number of countably compact group topologies on a free Abelian group,
Topology Appl. 98 (1999), 345–353.
[44] A. H. Tomita, Two countably compact topological groups: One of size ℵω and the other ofweight ℵω , Proc. Amer. Math. Soc., to appear.

HEREDITARILY SEPARABLE GROUP TOPOLOGIES(D. Dikranjan)
Dipartimento di Matematica e Informatica
Università di Udine
Via delle Scienze 206, 33100 Udine
Italy
E-mail address: dikranja@dimi.uniud.it(D. Shakhmatov)
Department of Mathematical Sciences, Faculty of Science
Ehime University, Matsuyama 790–8577
JapanE-mail address: dmitri@dpc.ehime-u.ac.jp

27

Proceedings of General Topology Symposium, p. 29 - p. 34, 2002

http://gt2002.h.kobe-u.ac.jp/

ON THE PROPERTY “SEP” OF PARTIAL ORDERINGSSAKAÉ FUCHINO

Abstract. A partial ordering P has the property SEP (or SEP(P )), if, fora sufficiently large regular cardinal χ, the family of elementary submodels ofH(χ) of cardinality ℵ1 with the property that [M ]ℵ0 ∩M is cofinal subset (withrespect to ⊆) of [M ]ℵ0 and P ∩ M ≤σ P , is cofinal (also with respect to ⊆)in [H(χ)]ℵ1 (see Proposition 2 below). We describe, in section 1, against
which historical background this notion came to be formulated. In section 2,
we introduce the property SEP and some other related properties of partial orderings P such as the weak Freese-Nation property and (ℵ1 , ℵ0 )-ideal property
and review some known results around these properties. In section 3 we givea sketch of a proof of the theorem asserting that the combinatorial principle
Princ introduced by S. Shelah does not imply SEP of hP(ω), ⊆i.

The following is based on author’s talk at General Topology Symposium 2002
held on 18–20, November 2002 at Kobe University, Japan. The property SEP to
be introduced in section 2 will be the main subject of this article. But since this isgoing to appear in the proceedings of the general topology symposium, let us begin
with a historical account explaining the connection to topology.
1. Historical background
Remember that a zero-dimensional Hausdorff space is also called a Boolean space— a topological space X is said to be zero-dimensional if closed and open (clopen)
subsets of X constitute a topological base of X.
It is well-known that there are contravariant functors between the category of
Boolean algebras and the category of Boolean spaces (Stone Duality Theorem). By
the duality, a Boolean algebra B is related to the space U lt(B) of all ultra-filters
on B with a canonical topology and a Boolean space X is related to the Boolean
algebra Clop(B) of clopen subsets of X partially ordered by ⊆ (see e.g. [16]).
One of the most important classes of Boolean spaces is that of the generalized
Cantor spaces κ2 which are the product spaces of κ copies of the discrete space
2 = {0, 1} with their Boolean algebraic dual being free Boolean algebras F r(κ)
with a free generator of size κ. A variety of classes of topological spaces which
are more or less similar to the generalized Cantor spaces, such as dyadic spaces,
Dugundji spaces and κ-metrizable spaces, have been studied extensively in theliterature (see e.g. the reference of [17] and [12]).
These classes of topological spaces have natural counterparts in the category of
Boolean algebras via Stone Duality Theorem (see [17], [12]):29

30

SAKAÉ FUCHINO

Boolean space which is:κ
2dyadic space
Dugundji spacecorresponding notion in Boolean algebra
F r(κ)
subalgebra of a free Boolean algebra
projective Boolean algebra

κ-metrizable space

Boolean algebra with Freese-Nation property(also called openly generated Boolean algebra)

The last line of the chart above is a result by Lutz Heindorf in [12] which can beformulated as follows:
Theorem 1.1. (L. Heindorf [12]) A Boolean algebra B is a Boolean algebraic dualof a κ-metrizable space if and only if there is a mapping f : B → [B]<ℵ0 such that(∗) For any a, b ∈ B, a ≤ b, there is c ∈ f (a) ∩ f (b) such that a ≤ c ≤ b.
A Boolean algebra with an f as above is said to have the Freese-Nation property after R. Freese and B. Nation who studied this property in connection withprojective lattices in [2]. Soon after this result, an interesting weakening of theFreese-Nation property was formulated in [12] and [9]: a Boolean algebra B is said
to have the weak Freese-Nation property (wFN) if there is a mapping f : B → [B]ℵ0
with the property (∗) as above. For a topological translation of the wFN see [12].
The following is an almost trivial but very useful characterization of Boolean
algebras with the wFN:
Theorem 1.2. (S. Fuchino, S. Koppelberg and S. Shelah [9]) A Boolean algebra
B has the wFN if and only if, for any sufficiently large regular κ and for any
M ≺ hH(χ), ∈i with B ∈ M and | M | = ℵ1 , we have B ∩ M ≤σ B.
Here H(χ) denotes the set consisting of all sets of hereditary of cardinality < χ.
For a partial ordering hP, ≤i and its subordering Q, Q is said to be a σ-suborderingof P (or Q ≤σ P ) if and only if for any p ∈ P , Q ¹ p = {q ∈ Q : q ≤ p} has acofinal subset of size ≤ ℵ0 and Q ↑ p = {q ∈ Q : q ≥ p} has a coinitial subset ofsize ≤ ℵ0 . Note that, in case of a Boolean algebra B and its subalgebra A, it isenough to check that every ideal in A of the form A ¹ b = {a ∈ A : a ≤ b} for someb ∈ B is countably generated to see that A is a σ-subalgebra of B.
Let us say that a partial ordering P has the wFN (notation: WFN(P )) if P
satisfies the property given in Theorem 1.2, i.e. if for any sufficiently large regularκ and for any M ≺ hH(χ), ∈i with P ∈ M and | M | = ℵ1 , we have P ∩ M ≤σ P .
An interesting point about the wFN is that also the Boolean algebra hP(ω), ⊆i
can have this property†. Under CH this is trivially so but also in a model of ZFC
obtained by adding Cohen reals e.g. to a model of V = L (see [9] and [10]).
In [8], it is shown that under the assumption‡ of the wFN of hP(ω), ⊆i, we canprove many of the combinatorial statements known to hold in Cohen models.
2. SEP and some other weakenings of the weak Freese-Nation
property
A. Dow and K.P. Hart defined the following weakening of the property WFN(P )of a partial ordering P in [1]: A partial ordering P has the (ℵ1 , ℵ0 )-Ideal Property(abbreviation: IDP(P )) if the following holds:†

In contrast, hP(ω), ⊆i never has the Freese-Nation property.
This assumption is also denoted as “WFN”.

‡

ON THE PROPERTY “SEP” OF PARTIAL ORDERINGS

31

IDP(P ): For any sufficiently large regular κ and for any M ≺ hH(χ), ∈i withP ∈ M and | M | = ℵ1 such that [M ]ℵ0 ∩ M is cofinal in [M ]ℵ0 , wehave P ∩ M ≤σ P .
The question if WFN(P ) is equivalent with IDP(P ) is a very delicate one:
Theorem 2.1. (S. Fuchino and L. Soukup [10]) For partial ordering P of cardinality < ℵω , WFN(P ) if and only if IDP(P ). If a very weak form of square principle
holds for cardinals of countable cofinality then WFN(P ) and IDP(P ) are equivalent
for arbitrary partial ordering P .
On the other hand, as is proved in [10], there is a partial ordering P with
¬WFN(P ) and IDP(P ) under (ℵω )ℵ0 = ℵω+1 and Chang’s conjecture (ℵω+1 , ℵω ) ³(ℵ1 , ℵ0 ).
It is shown in [1](independently from [8]) that most of the results obtained in[8] under the wFN of P(ω) can be already proved under the weaker assumption of
IDP of P(ω).
In [13], I. Juhász and K. Kunen introduced a property which they called SEP.
For a Boolean algebra B let SEP(B) be the following assertion:
SEP(B): For every sufficiently large regular cardinal χ, the set of those M ≺H(χ) satisfying the following conditions (0)∼(2) is cofinal in [H(χ)]ℵ1 :(0) B ∈ M and | M | = ℵ1 ;(1) [M ]ℵ0 ∩ M is cofinal in [M ]ℵ0 ;(2) B ∩ M ≤sep B.
Here, for a Boolean algebra A and its subalgebra B, B ≤sep A if and only if for alla ∈ A and K ∈ [B ¹ a]ℵ1 , there is b ∈ B ¹ a such that | K ∩ B ¹ b | = ℵ1 whereB ¹ a denotes as before the ideal {d ∈ B : d ≤ a}.
SEP has a characterization which put it in line with wFN and IDP. This follows
from the next lemma. ( S. Fuchino and S. Geschke [6]) Suppose that A is a Boolean
algebra and B its subalgebra. Then
(1) B ≤σ A implies B ≤sep A.
(2) If | B | ≤ ℵ1 then B ≤σ A implies B ≤sep Aℵ1
Proof. (1): Suppose that B ≤σ A. For a ∈ A
S and K ∈ [B ¹ a] . Let {bn :n ∈ ω} be a cofinal subset of B ¹ a. Then K = n∈ω (K ∩ B ¹ bn ). Hence one ofK ∩ B ¹ bn , n ∈ ω must be uncountable.
(2): Let B = {bα : α < ω1 }. Assume that B ≤sep A but B is not a σ-subalgebraof A. Then there is an a ∈ A such that B ¹ a is not countably generated. Letcα ∈ A ¹ b be taken inductively so that cα is not in the ideal in B generated by
Gα = {cβ : β < α} ∪ {bβ : β < α, bβ ≤ a}.
This is possible since each Gα ⊆ B ¹ a is countable. Let K = {cα : α < ω1 }. By
assumption there is some β0 < ω1 such that bβ0 ≤ a and K ∩B ¹ bβ0 is uncountable.
Then there is some β0 < α < ω1 such that cα ∈ K ∩ B ¹ bβ0 . But, since bβ0 < a,(Lemma 0)bβ0 ∈ Gα and so cα 6≤ bβ0 . This is a contradiction.
(S. Fuchino and S. Geschke [6]) For a Boolean algebra B, SEP(B) if and only if
the following holds:

32

SAKAÉ FUCHINO

For every sufficiently large regular cardinal χ, the set of those M ≺ H(χ)
satisfying the following conditions (0)∼(2) is cofinal in [H(χ)]ℵ1 :(0) B ∈ M and | M | = ℵ1 ;(1) [M ]ℵ0 ∩ M is cofinal in [M ]ℵ0 ;(2’) B ∩ M ≤σ B.
Proof. By Lemma 2.

(Proposition 0)

Note that the characterization of SEP is applicable for partial orderings as well.
Hence we shall say SEP(P ) for a partial ordering P if, for every sufficiently large
regular cardinal χ, the set of those M ≺ H(χ) satisfying the following conditions(0)∼(2’) is cofinal in [H(χ)]ℵ1 : (0) P ∈ M and | M | = ℵ1 ; (1) [M ]ℵ0 ∩ M is cofinal
in [M ]ℵ0 ; (2’) P ∩ M ≤σ P .
Clearly we have
WFN(P ) ⇒ IDP(P ) ⇒ SEP(P )for all partial ordering P . In [6] it is proved (in ZFC without any additional assumptions) that there exists a Boolean algebra B such that ¬IDP(B) and SEP(B).
3. Princ does not imply SEP
Let us denote by WFN, IDP and SEP the combinatorial statements WFN(P(ω)),
IDP(P(ω)) and SEP(P(ω)) respectively where P(ω) is seen here as before as thepartial ordering hP(ω), ⊆i. It is shown in [7] that WFN 6⇐ IDP is consistent with
ZFC modulo some quite large cardinal. The consistency of IDP 6⇐ SEP (without
any large cardinal) is proved in [6]. Thus we have:
WFN

⇒
6⇐

IDP

⇒
6⇐

SEP

SEP is still strong enough to drive most of the results proved in [8] under theassumption of WFN. Most of the proofs in [8] can be easily modified to a proof
under SEP. With one exception: The proof of a = ℵ1 under WFN in [8] works
without any problem under IDP but it seems that the proof cannot be recast forthe proof under SEP. What we have right now is the following slightly weaker result:
Theorem 3.1. (S. Fuchino and S. Geschke [6]) Assume ¤ω1 . Then SEP implies
a = ℵ1 .
There are some other combinatorial principles which are also related to theseprinciples. One of them is introduced by S. Shelah and called Princ:
Princ: For every sufficiently large regular cardinal χ, the set of those M ≺H(χ) satisfying the following conditions (I)∼(III) is cofinal in [H(χ)]ℵ1 :(I) | M | = ℵ1 ;(II) ω2 ∩ M ∈ ω2 ;ℵ(III) For all a ∈ P(ω) there is X ∈ [P(ω)] 1 ∩ M such that X ∩ P(a)is cofinal in P(a) ∩ M .
SEP implies Princ. Proof. Assume SEP and suppose that χ is sufficiently large
and M ≺ H(χ) is such that | M | = ℵ1 , [M ]ℵ0 ∩ M is cofinal in [M ]ℵ0 and P(ω) ∩M ≤σ P(ω).
Note that, by assumption, there are cofinally may such M ’s in [H(χ)]ℵ1 . Hence
it is enough to show that M as above satisfies (I), (II) and (III) in the definition of
Princ.

ON THE PROPERTY “SEP” OF PARTIAL ORDERINGS

33

M |= (I) is clear. For (II), note that ω1 ⊆ M . Suppose α ∈ ω2 ∩ M . Then, by
elementarity, there is f ∈ M such that f : ω1 → α and f is surjective. It follows
that α = f 00 ω1 ⊆ M . Hence ω2 ∩ M is an initial segment of ω2 of cardinality ℵ1and thus an element of ω2 .
To show M |= (III), let a ∈ P(ω). Then there is a countable X 0 ⊆ P(a) ∩ Mℵsuch that X 0 is cofinal in P(a) ∩ M . Let X 00 ∈ [P(ω)] 0 ∩ M be such that X 0 ⊆ X 00 .
ℵ
By elementarity, there is some X ∈ [P(ω)] 1 ∩ M with X 00 ⊆ X. This X is clearly
as in (III) for our a.

(Lemma 0)

To close this section, we shall give a sketch of the proof the the converse of Lemma
0does not hold. The following Theorem is due to Stefan Geschke:
Theorem 3.2. Suppose that V |= CH, Q is a partial ordering satisfying the c.c.c.
and there is a partial ordering P of cardinality ≤ ℵ1 such that Q is a finite support
product of copies of P . Then k– Q “ Princ ”.
Theorem 3.3. ¬SEP + Princ is consistent with ZFC.
Proof. Start from a model V of CH. In V , let P be the partial ordering for adding
a single random real and let Q be a finite support product of κ > ℵ1 copies of P .
Then k– Q “ ¬SEP ” by Lemma 3.1.6 in [11] and Theorem 8.1 in [6]. On the other
hand k– Q “ Princ ” by Theorem 3.2.

(Theorem 0)

4. Some more principles
The following diagram summarizes known implications among combinatorial
principles discussed in the previous sections together with some other combinatorial principles from [14], [15] and [5]:

WFN
[7]

6⇑ ⇓

IDP

CH∗
-45⇒

[13]

[6]

6⇑ ⇓

SEP
6⇑ ⇓ (†)

(*)

Princ
6⇑ ⇓ (**)[13]

IP(ω2 )⇓
HP(ω2 )
45⇐ 456⇒ [13]sC (ω2 )(†) By Lemma 3.
(*) By Theorem 3.3.
(**) By S. Shelah in an unpublished note.

34

SAKAÉ FUCHINO

References[1] A. Dow and K.P. Hart, Applications of another characterization of βN \ N, Topology and its
Applications, 122, 1-2, 105–133 (2002)[2] R. Freese and J. B. Nation, Projective lattices, Pacific Journal of Math. 75 (1978), 93–106.
[3] M. Foreman, M. Magidor, and S. Shelah, Martin’s Maximum, saturated ideals and nonregular ultrafilters. Part I. AM 127 (1988), 1-47.
[4] S. Fuchino, Some remarks on openly generated Boolean algebras, Journal of Symbolic Logic,
Vol. 59 (1994), 302–310.
[5] S. Fuchino and J. Brendle, Coloring ordinals by reals, preprint.
[6] S. Fuchino and S. Geschke, Remarks on a paper by Juhász and Kunen, preprint.
[7] S. Fuchino, S. Geschke, S. Shelah and L. Soukup, On the weak Freese-Nation property ofcomplete Boolean algebras, Annals of Pure and Applied Logic 110, No.1-3 (2001), 89–105.
[8] S. Fuchino, S. Geschke and L. Soukup, The weak Freese-Nation property of P(ω), Archive ofMathematical Logic 40, No.6 (2001), 425–435.
[9] S. Fuchino, S. Koppelberg and S. Shelah, Partial orderings with the weak Freese-Nation
property, Annals of Pure and Applied Logic, 80 (1996).
[10] S. Fuchino and L. Soukup, More set theory around the weak Freese-Nation property, Fundamenta Mathematicae 154 (1997), 159–176.
[11] S. Geschke, On σ-filtered Boolean algebras, Ph.D. Thesis, Berlin (1999).
[12] Lutz Heindorf and Leonid B. Shapiro, Nearly Projective Boolean Algebras, Springer Lecture
Notes in Mathematics Vol. 1596 (1994).
[13] I. Juhász and K. Kunen, The Power Set of ω, Elementary submodels and weakenings of CH,
Fundamenta Mathematicae 170 (2001), 257–265.
[14] I. Juhász, L. Soukup, and Z. Szentmiklóssy, Combinatorial principles from adding Cohen
reals, in Logic Colloquium 95, Haifa, Israel, Springer (1998), 79–103.
[15] I. Juhász, L. Soukup, and Z. Szentmiklóssy, What left from CH if we add Cohen reals,
Topology Appl. (1998), 165–174.
[16] S. Koppelberg, General Theory of Boolean Algebras, Vol.1 of: D. Monk with R. Bonnet (ed.),Handbook of Boolean Algebras, North-Holland (1989).
[17] S. Koppelberg, Projective algebras, in: D. Monk with R. Bonnet (ed.), Handbook of Boolean
Algebras, Vol.3, North-Holland, (1989), 741–774.
[18] K. Kunen, Set-Theory, North-Holland (1980).
Department of Natural Science and Mathematics, Chubu University. Kasugai Aichi
487-8501 JAPANE-mail address: fuchino@isc.chubu.ac.jp

Proceedings of General Topology Symposium, p. 35 - p. 44, 2002

http://gt2002.h.kobe-u.ac.jp/

SELECTIONS AND SANDWICH-LIKE PROPERTIES VIASEMI-CONTINUOUS BANACH-VALUED FUNCTIONSVALENTIN GUTEV, HARUTO OHTA, AND KAORI YAMAZAKI

1. Introduction
Since the paper will be published in the J. Math. Soc. Math. soon, we omit allof the proofs and some lemmas. Throughout this paper, by a space we mean a nonempty T1 -space. Our investigation was motivated by the following two theorems;
the former was proved by Katětov [14, 15] and Tong [29], and the latter was provedby Kandô [13] and Nedev [23]:Theorem 1.1 (Katětov-Tong’s insertion theorem). A space X is normal if andonly if for every two functions g, h : X → R such that g is upper semi-continuous,
h is lower semi-continuous and g ≤ h, there exists a continuous function f : X → R
such that g ≤ f ≤ h.
For a Banach space Y , let Fc (Y ) (resp., Cc (Y )) denote the set of all non-empty
closed (resp., non-empty compact) convex sets in Y . A map f : X → Y is called aselection of a mapping φ : X → Fc (Y ) if f (x) ∈ φ(x) for every x ∈ X.
Theorem 1.2 (Kandô-Nedev’s selection theorem). Let λ be an infinite cardinal.
Then, the following conditions on a space X are equivalent:(1) Every point-finite open cover U of X, with |U| ≤ λ, is normal.
(2) For every Banach space Y , with w(Y ) ≤ λ, every lower semi-continuous
mapping φ : X → Cc (Y ) admits a continuous selection.
(3) Every lower semi-continuous mapping φ : X → Cc (`1 (λ)) admits a continuous selection.
Theorem 1.2 can be regarded as an essential part of Michael’s selection theorem[19, Theorem 3.20 ] (see, also, [2]) asserting that a space X is λ-collectionwise normal
if and only if X satisfies the condition (2) with Cc (Y ) replaced by Cc (Y ) ∪ {Y }.
For a space Y , let C0 (Y ) denote the Banach space of all real-valued continuousfunctions s on Y such that for each ε > 0 the set {y ∈ Y : |s(y)| ≥ ε} is compact,
where the linear operations are defined pointwise and ksk = supy∈Y |s(y)| for eachs ∈ C0 (Y ). In particular, we use c0 (λ) to denote the space C0 (Y ), where Y is thediscrete space of cardinality λ, i.e. c0 (λ) is the Banach space consisting of all pointss ∈ Rλ such that the set {α < λ : |s(α)| ≥ ε} is finite for each ε > 0.
In this paper, we introduce lower and upper semi-continuity of a map to C0 (Y ).
We prove that if the space R in Theorem 1.1 is replaced by c0 (λ), then the resulting
statement is equivalent to the conditions listed in Theorem 1.2, see Theorem 3.1.
Thus, insertions and selections are connected via the space c0 (λ). As a result, we35

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V. GUTEV, H. OHTA, AND K. YAMAZAKI

obtain several sandwich-like analogues to selection theorems as well as selection
theorems corresponding to sandwich-like properties, see Section 4.
For set-valued mappings ϕ and ψ defined on a space X, we say that ϕ is a setvalued selection of ψ, or ψ is an expansion of ϕ, if ϕ(x) ⊆ ψ(x) for each x ∈ X.
Let C(Y ) denote the set of all non-empty compact sets in a space Y . In [23] Nedev
has characterized several paracompact-like properties by the existence of set-valued
selections of C(Y )-valued mappings for completely metrizable spaces Y . In contrast
to this, we characterize expandability and almost expandability in the sense of[16, 27] by insertion of c0 (λ)-valued maps, and by the existence of expansions ofC(Y )-valued mappings for completely metrizable spaces Y , see Section 5.
We often consider two kinds of maps in the same statement, i.e., a single-valued
map to a space Y and a set-valued map to a hyperspace of Y . To distinguish them,
we use the term map for the former one and the term mapping for the latter one.
As usual, a cardinal is identified with the initial ordinal and an ordinal is the setof all smaller ordinals. The cardinality of a set A is denoted by |A|. Let ω denotethe first infinite cardinal and N the set of non-negative integers. Other terms andnotation will be used as in [8].
2. Semi-continuous C0 (Y )-valued functions and compact sets
In this section, X and Y denote arbitrary spaces and λ stands for a cardinal. Fora real-valued function f : X → R and r ∈ R, let L(f, r) = {x ∈ X : f (x) > r} andU (f, r) = {x ∈ X : f (x) < r}. Recall that a function f : X → R is lower (resp.,
upper ) semi-continuous if L(f, r) (resp., U (f, r)) is open in X for each r ∈ R. Now,
we extend these notions to C0 (Y )-valued maps as follows:
Definition 2.1. A map f : X → C0 (Y ) is lower (resp., upper ) semi-continuous if
for every x ∈ X and every ε > 0, there is a neighbourhood G of x in X such that if
x0 ∈ G, then f (x0 )(y) > f (x)(y) − ε (resp., f (x0 )(y) < f (x)(y) + ε) for each y ∈ Y .
With every map f : X → C0 (Y ) we associate another one −f : X → C0 (Y )defined by (−f )(x)(y) = −f (x)(y) for each x ∈ X and each y ∈ Y . The firstlemma is a direct consequence of the definition.
Lemma 2.2. A map f : X → C0 (Y ) is continuous if and only if it is both lower
and upper semi-continuous. A map f : X → C0 (Y ) is lower semi-continuous if andonly if the map −f is upper semi-continuous.
The following three lemmas concern only the case of c0 (λ). For each α < λ, let
πα : Rλ → R denote the α-th projection, i.e. πα (s) = s(α) for s ∈ Rλ .
Lemma 2.3. For a map f : X → c0 (λ), the following are valid :(1) f is lower semi-continuous if and only if πα ◦ f is lower semi-continuousfor each α < λ, and {U (πα ◦ f, −ε) : α < λ} is locally finite in X for eachε > 0.
(2) f is upper semi-continuous if and only if πα ◦ f is upper semi-continuousfor each α < λ, and {L(πα ◦ f, ε) : α < λ} is locally finite in X for eachε > 0.
Lemma 2.4. Let f : X → Rλ be a map. Then, f [X] ⊆ c0 (λ) if and only if both{L(πα ◦ f, ε) : α < λ} and {U (πα ◦ f, −ε) : α < λ} are point-finite in X for eachε > 0.

SELECTIONS AND SANDWICH-LIKE PROPERTIES37

For s ∈ C0 (Y ) and ε > 0, let B(s, ε) = {t ∈ C0 (Y ) : ks − tk < ε}. Fors, t ∈ C0 (Y ), we write s ≤ t if s(y) ≤ t(y) for each y ∈ Y . Further, if s ≤ t, thenwe define [s, t] = {u ∈ RY : s ≤ u ≤ t}. Obviously, [s, t] is a closed convex subset
of C0 (Y ). In the case of c0 (λ), we have a stronger result:Lemma 2.5. For every s, t ∈ c0 (λ), with s ≤ t, the subspace topology σ on [s, t]
coincides with the subspace topology induced from the product topology τ on Rλ .
Hence, in particular, [s, t] is a compact convex subset of c0 (λ).
We now recall the definitions of upper and lower semi-continuity of set-valued
mappings. Let φ : X → S be a set-valued mapping, where S is a family of nonempty subsets of a space Y . For a subset U ⊆ Y , let φ−1 [U ] = {x ∈ X : φ(x) ∩ U 6=∅} and φ# [U ] = {x ∈ X : φ(x) ⊆ U }. The mapping φ : X → S is called lower(resp., upper ) semi-continuous if φ−1 [U ] (resp., φ# [U ]) is open in X for every
open set U in Y . Also, φ is called continuous if it is both lower and upper semicontinuous.
For maps g, h : X → C0 (Y ), we shall write g ≤ h if g(x) ≤ h(x) for everyx ∈ X. With every two such maps we associate a set-valued mapping [g, h] : X →Fc (C0 (Y )) defined by [g, h](x) = [g(x), h(x)] for x ∈ X. Also, we associate two
mappings [g, +∞) and (−∞, h] from X to Fc (C0 (Y )) by [g, +∞)(x) = {s ∈ C0 (Y ) :s ≥ g(x)} and (−∞, h](x) = {s ∈ C0 (Y ) : s ≤ h(x)} for x ∈ X, respectively.
Finally, for S ⊆ C0 (YS) and ε > 0, let B(S, ε) denote the ε-neighbourhood of S in
C0 (Y ), i.e. B(S, ε) = s∈S B(s, ε).
Lemma 2.6. Let g, h : X → C0 (Y ) be maps such that g ≤ h.
(1) If g is upper semi-continuous, then [g, +∞) is lower semi-continuous.
(2) If h is lower semi-continuous, then (−∞, h] is lower semi-continuous.
(3) If g is upper semi-continuous and h is lower semi-continuous, then the
mapping [g, h] is lower semi-continuous.
(4) If g is lower semi-continuous, h is upper semi-continuous and Y is discrete,then the mapping [g, h] is upper semi-continuous.
For a non-empty bounded set K ⊆ C0 (Y ), we define points sup K and inf K
of RY by (sup K)(y) = sup{s(y) : s ∈ K} and (inf K)(y) = inf{s(y) : s ∈ K},
respectively, for each y ∈ Y .
Lemma 2.7. If K is a non-empty compact set in C0 (Y ), then sup K ∈ C0 (Y ) andinf K ∈ C0 (Y ). Hence, K ⊆ [inf K, sup K].
For a mapping φ : X → C(C0 (Y )), we define single-valued maps sup φ : X →C0 (Y ) and inf φ : X → C0 (Y ) by (sup φ)(x) = sup φ(x) and (inf φ)(x) = inf φ(x),respectively, for each x ∈ X.
Lemma 2.8. Let φ : X → C(C0 (Y )) be a mapping.
(1) If φ is lower semi-continuous, then sup φ is lower semi-continuous and inf φis upper semi-continuous.
(2) If φ is upper semi-continuous, then sup φ is upper semi-continuous andinf φ is lower semi-continuous.
3. Extension of Theorem 1.2
For two families F and G of subsets of a space X, we call G an expansion of F
if there exists a bijection G : F → G such that F ⊆ G(F ) for each F ∈ F. An

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V. GUTEV, H. OHTA, AND K. YAMAZAKI

open expansion is an expansion consisting of open sets. For real-valued functions
fα , α < λ, on a space X, let 4α<λ fα denote the map f : X → Rλ such thatπα ◦ f = fα for each α < λ.
In this section, we find a natural relationship between insertions and selections
by proving the following theorem which extends Theorem 1.2. The equivalence of(1) and (2) is due to Kandô [13] and Nedev [23] as was stated in the introduction.
Theorem 3.1. For an infinite cardinal λ, the following conditions on a space X
are equivalent:(1) Every point-finite open cover U of X, with |U| ≤ λ, is normal.
(2) For every Banach space Y , with w(Y ) ≤ λ, every lower semi-continuous
mapping ϕ : X → Cc (Y ) admits a continuous selection.
(3) Every lower semi-continuous mapping ϕ : X → Cc (c0 (λ)) admits a continuous selection.
(4) For every two maps g, h : X → c0 (λ) such that g is upper semi-continuous,
h is lower semi-continuous and g ≤ h, there exists a continuous map f :X → c0 (λ) such that g ≤ f ≤ h.
(5) For every two maps g, h : X → c0 (λ) such that g is upper semi-continuous,
h is lower semi-continuous and g ≤ h, there exist a lower semi-continuous
map f` : X → c0 (λ) and an upper semi-continuous map fu : X → c0 (λ)
such that g ≤ f` ≤ fu ≤ h.
(6) X is normal, and every locally finite family F of closed sets in X, with
|F| ≤ λ, has a locally finite open expansion provided it has a point-finite
open expansion.
(7) Every discrete family F of closed sets in X, with |F| ≤ λ, has a disjoint
open expansion provided it has a point-finite open expansion.
Remark 3.2. The following conditions (8) and (9) are also equivalent to the conditions listed in Theorem 3.1. For two mappings ϕ, ψ : X → C(Y ), we write ϕ ⊆ ψ if
ϕ(x) ⊆ ψ(x) for each x ∈ X.
(8) For every metrizable space Y , with w(Y ) ≤ λ, and every lower semicontinuous mapping φ : X → C(Y ), there exist a lower semi-continuous
mapping ϕ : X → C(Y ) and an upper semi-continuous mapping ψ : X →C(Y ) such that ϕ ⊆ ψ ⊆ φ.
(9) There exist a space Y and a disjoint family G of non-empty open sets in Y ,with |G| = λ, such that for every lower semi-continuous mapping φ : X →C(Y ), there exists an upper semi-continuous mapping ψ : X → C(Y ) suchthat ψ ⊆ φ.
The equivalence of (1) and (8) was proved by Nedev in [23, Theorem 3], while(8) ⇒ (9) is obvious. To show that (9) ⇒ (7), let F be a discrete family of closed
sets in X, with |F| ≤ λ, and U = {U (F ) : F ∈ F } be a point-finite open expansionof F. We may assume that U covers X and U (F ) ∩ F 0 = ∅ whenever F 6= F 0 . On
the other hand, there exists a disjoint family G = {G(F ) : F ∈ F} of non-empty
open sets in Y . Fix a point yF ∈ G(F ) for each F ∈ F and define φ : X → C(Y )by φ(x) = {yF : x ∈ U (F ), F ∈ F} for x ∈ X. Then, φ is lower semi-continuous
because U is an open cover of X. Hence, by (9), there exists an upper semicontinuous mapping ψ : X → C(Y ) such that ψ ⊆ φ. Let V (F ) = ψ # [G(F )] foreach F ∈ F. Then {V (F ) : F ∈ F} is a disjoint open expansion of F.

SELECTIONS AND SANDWICH-LIKE PROPERTIES

39

Let λ-PN be the class of all spaces satisfying one of (and hence, all of) the
conditions listed in Theorem 3.1. Define the class PN by X ∈ PN if and only
if X ∈ λ-PN for every cardinal λ. Then, PN is included in the class N of all
normal spaces and contains the class CN of all collectionwise normal spaces, i.e.
CN ⊆ PN ⊆ N . Michael [18] has shown that both inclusions are proper by giving
the examples which we now sketch below:
The example showing that PN 6= CN is the standard Bing’s example (cf. [8,cExample 5.1.23]). The product space X = D2 of the discrete space D = {0, 1}
contains a discrete subspace M , with |M | = c. Bing’s space Z is obtained from thespace X by making all points of X\M isolated. It is known that Z ∈ N \CN . Noticethat every point-finite family of non-empty open sets in X is at most countable;
this follows from the fact that the Šanin number of X is countable (cf. [8, 2.7.11, p.
116]). Hence, it follows that Z ∈ PN . Next, consider the subspace Y = M ∪ D ofZ, where D = {x ∈ X : {α < 2c : x(α) 6= 0} is finite}. Michael [18] has shown thatthe space Y is normal metacompact but not paracompact. Hence, Y ∈ N \ PN
because every metacompact space in PN must be paracompact.
Since the space Y = M ∪ D is closed in Bing’s space Z, the example above also
shows that the class PN is not closed under taking closed subspaces unlike N and
CN . From this fact, it is natural to ask whether a space X is in CN if every closed
subspace of X is in PN . Now, we show that the answer is negative if there existsa Q-set. To this end, let us recall that a subset A of the real line R is called aQ-set if A is uncountable and every subset of A is a Gδ -set in A with respect to thesubspace topology on A inherited from the usual topology on R. It is known thatevery uncountable subset A ⊆ R, with |A| < c, is a Q-set under assuming Martin’s
axiom and the negation of the continuum hypothesis (see [20] for details).
Example 3.3. If there exists a Q-set in R, then there exists a perfectly normal
space X such that every subspace is in PN but X 6∈ CN .
Problem 3.4. Does there exist an example in ZFC of a space X 6∈ CN such thatevery closed subspace of X is in PN ?
4. Sandwich-like characterizations of paracompact-like properties
A space X is called λ-collectionwise normal if every discrete family F of closed
sets in X, with |F| ≤ λ, has a discrete open expansion. In what follows, for a
Banach space Y , we put Cc0 (Y ) = Cc (Y ) ∪ {Y }.
Our first result is an insertion-like theorem which characterizes λ-collectionwise
normality.
Theorem 4.1. Let λ be an infinite cardinal. For a space X the following conditions
are equivalent:(1) X is λ-collectionwise normal.
(2) For every Banach space Y , with w(Y ) ≤ λ, every lower semi-continuous
mapping ϕ : X → Cc0 (Y ) has a continuous selection.
(3) Every lower semi-continuous mapping φ : X → Cc0 (c0 (λ)) has a continuous
selection.
(4) For every closed subspace A of X and for every two maps g, h : A → c0 (λ)
such that g is upper semi-continuous, h is lower semi-continuous and g ≤ h,there exists a continuous map f : X → c0 (λ) such that g ≤ f |A ≤ h.

40

V. GUTEV, H. OHTA, AND K. YAMAZAKI

Our next result is a characterization of countably paracompact and λ-collectionwise
normal spaces. To state our characterization of countably paracompact and λcollectionwise normal spaces, we need also some terminology about Banach spaces.
Let Y be a space and let e : Y → Rλ be a map. Then we define eα = πα ◦ e,
where πα : Rλ → R is the projection to the α-th factor of Rλ , for each α < λ.
Thus, we have e = 4{eα : α < λ}.
Suppose that Y is a Banach space. Let us recall that a sequence {en ∈ Y :n < ω}
P is a Schauder basis for Y if any point y ∈ Y has an unique representationy =coordinates)yn ∈ R, n < ω. Here, y =n<ω yn en for some scalars° (i.e.,°PP°°n<ω yn en means that limn→∞ °y −k≤n yk ek ° = 0, where k.k is the norm of Y .
Note that any Schauder basis {en ∈ Y : n < ω} for a Banach space Y definesa natural linear continuous injection e : Y → Rω , see [3, Exercise III.14.10] andω
[26, Theorem
P 3.1]. Namely, one may define e : Y → R by en (y) = yn , n < ω,
where y = n<ω yn en ∈ Y . It should be mentioned that, with respect to this map
e = 4{en : e < ω}, we have en (en ) = 1 and em (en ) = 0 for m 6= n. Motivated bythis, we shall say that a map e : Y → Rλ is a generalized Schauder basis for a Banach
space Y if it is a continuous linear injection such that, whenever y ∈ Y and α < λ,there is a point yα ∈ Y , with eβ (yα ) = eβ (y) if β = α and eβ (yα ) = 0 otherwise.
Clearly, the natural linear injection e : Y → Rω determined by a Schauder basis
for Y is a generalized Schauder basis but the converse does not hold. For instance,
consider the Banach space `∞ of bounded sequences. Then the natural injectione : `∞ → Rω is a generalized Schauder basis but the space `∞ does not have a
Schauder one since it is not separable.
The generalized Schauder basises will be used in the following special situation.
Definition 4.2. We shall say that a generalized Schauder basis e : Y → Rλ for a
Banach space Y is a c0 (λ)-basis for Y if e[Y ] ⊂ c0 (λ) and it is continuous as a map
from Y to c0 (λ). Also, we shall say that Y is a generalized c0 (λ)-space if it is a
Banach space, with w(Y ) ≤ λ, which has a c0 (λ)-basis.
Note that c0 (λ) is a generalized c0 (λ)-space. Also, every Euclidean space is a
generalized c0 (λ)-space for every infinite cardinal λ. Finally, the Banach spaces`p (λ), for p ≥ 1, are another important example of generalized c0 (λ)-spaces.In
what follows, for a convex set K of a Banach space Y , we consider an weak convex
interior wci(K) of K defined by
wci(K) = {x ∈ K : x = δx1 + (1 − δ)x2 for some x1 , x2 ∈ K \ {x} and 0 < δ < 1}.
Also, for s, t ∈ Rλ , we shall write s < t if s ≤ t and s(α) < t(α) for some α < λ.
Finally, for maps g, h : X → Rλ , we write g < h if g(x) < h(x) for every x ∈ X.
Theorem 4.3. Let λ be an infinite cardinal. For a space X the following conditions
are equivalent:(1) X is countably paracompact and λ-collectionwise normal.
(2) Whenever Y is a generalized c0 (λ)-space and φ : X → Cc0 (Y ) is a lower
semi-continuous mapping such that |φ(x)| > 1 for every x ∈ X, there exists
a continuous map f : X → Y such that f (x) ∈ wci(φ(x)) for all x ∈ X.

SELECTIONS AND SANDWICH-LIKE PROPERTIES

41(3) For every lower semi-continuous mapping φ : X → Cc0 (c0 (λ)), with |φ(x)| >1 for every x ∈ X, there exists a continuous map f : X → c0 (λ) such thatf (x) ∈ wci(φ(x)) for all x ∈ X.
(4) For every closed subspace A of X and for every two maps g, h : A → c0 (λ)
such that g is upper semi-continuous, h is lower semi-continuous and g < h,there exists a continuous map f : X → c0 (λ) such that g < f |A < h.
From one hand, Theorem 4.3 might be read as a possible extension of the DowkerKatětov characterization of countably paracompact normal spaces [5, 14], see also[4]. From another hand, Theorem 4.3 should be compared with Michael’s characterization [19, Theorem 3.1000 ] of perfectly normal spaces by selections avoiding
supporting points of convex sets. More precisely, in the Michael’s terminology [19],if Y is a Banach space and K ∈ Fc (Y ), then a supporting set of K is a closed
convex subset S of K, S 6= K, such that if an interior point of a segment in K isin S, then the whole segment is in S. The set of all elements of K which are notin any supporting set of K is denoted by I(K) (suggesting “Inside of K”). Finally,
as in [19], one may consider©
ª
D(Y ) = B ∈ 2Y : B is convex and I(clY B) ⊆ B .
It is well known (see [19]) that Fc (Y ) ⊂ D(Y ); that every convex B ∈ 2Y witha non-empty interior belongs to D(Y ); and that every finite-dimensional convex
B ∈ 2Y belongs to D(Y ).
As for our weak convex interior, it is clear that I(K) ⊆ wci(K) for every K ∈Fc (Y ) but the converse is not true. In fact, the Michael’s [19, Theorem 3.1000 ] statesthat a space X is perfectly normal if and only if for every separable Banach space
Y , every lower semi-continuous φ : X → D(Y ) has a continuous selection.
Our next result present another possible characterization of perfectly normal
spaces in terms of selections.
Theorem 4.4. Let λ be an infinite cardinal. For a space X the following conditions
are equivalent:(1) X is perfectly normal and λ-collectionwise normal.
(2) Whenever Y is a generalized c0 (λ)-space, every lower semi-continuous mapping φ : X → Cc0 (Y ) has a continuous selection f such that f (x) ∈ wci(φ(x))
for every x ∈ X with |φ(x)| > 1.
(3) Every lower semi-continuous mapping φ : X → Cc0 (c0 (λ)) has a continuous
selection f such that f (x) ∈ wci(φ(x)) for every x ∈ X with |φ(x)| > 1.
(4) For every closed subspace A of X and for every two maps g, h : A → c0 (λ)
such that g is upper semi-continuous, h is lower semi-continuous and g ≤ h,there exists a continuous map f : X → c0 (λ) such that g ≤ f |A ≤ h andg(x) < f (x) < h(x) whenever x ∈ A with g(x) < h(x).
Returning back to Theorem 4.3, a word should be said about condition (2) ofthis theorem. In fact, the reader may wonder if this condition holds for all Banach
spaces. The authors do not know if this is true, which suggests the following natural
question.
Problem 4.5. Let X be a countably paracompact and λ-collectionwise normal
space for some infinite cardinal λ, Y be a Banach space with w(Y ) ≤ λ, andφ : X → Cc0 (Y ) be lower semi-continuous such that |φ(x)| > 1 for every x ∈ X.

42

V. GUTEV, H. OHTA, AND K. YAMAZAKI

Does there exist a continuous map f : X → Y such that f (x) ∈ wci(φ(x)) for everyx ∈ X?
5. Characterizations of expandable spaces
Let λ be an infinite cardinal. A space X is called λ-expandable (resp., almost
λ-expandable) if every locally finite family F of closed sets in X, with |F| ≤ λ,has a locally finite (resp., point-finite) open expansion (cf. [16, 27]). We state theresults, then proceed to the proofs.
Theorem 5.1. For an infinite cardinal λ, the following conditions on a space X
are equivalent:(1) X is λ-expandable.
(2) For every completely metrizable space Y , with w(Y ) ≤ λ, and every upper semi-continuous mapping φ : X → C(Y ), there exist two mappingsϕ, ψ : X → C(Y ) such that ϕ is lower semi-continuous, ψ is upper semicontinuous and φ ⊆ ϕ ⊆ ψ.
(3) There exists a space Y and a locally finite family G of non-empty open setsin Y , with |G| = λ, such that for every upper semi-continuous mapping
φ : X → C(Y ), there exist two mappings ϕ, ψ : X → C(Y ) such that ϕ islower semi-continuous, ψ is upper semi-continuous and φ ⊆ ϕ ⊆ ψ.
(4) For every upper semi-continuous map f : X → c0 (λ), there exist two maps
g, h : X → c0 (λ) such that g is lower semi-continuous, h is upper semicontinuous and f ≤ g ≤ h.
Theorem 5.2. For an infinite cardinal λ, the following conditions on a space X
are equivalent:(1) X is almost λ-expandable.
(2) For every completely metrizable space Y , with w(Y ) ≤ λ, and every upper semi-continuous mapping φ : X → C(Y ), there exists a lower semicontinuous mapping ϕ : X → C(Y ) such that φ ⊆ ϕ.
(3) There exists a space Y and a locally finite family G of non-empty open
sets in Y , with |G| = λ, such that for every upper semi-continuous mapping
φ : X → C(Y ), there exists a lower semi-continuous mapping ϕ : X → C(Y )such that φ ⊆ ϕ.
(4) For every upper semi-continuous map f : X → c0 (λ), there exists a lower
semi-continuous map g : X → c0 (λ) such that f ≤ g.
Miyazaki [21] has proven the equivalence (1) and (2) in Theorem 5.1 assumingthat X is normal, and has shown that every metacompact space satisfies (2) in
Theorem 5.2.
It is known ([16]) that a space X is ω-expandable if and only if it is countably
paracompact. Hence, by the definitions, a space X is λ-collectionwise normal andcountably paracompact if and only if X satisfies one of the following two conditions:(i) X is λ-expandable and X ∈ λ-PN ; (ii) X is almost λ-expandable and X ∈λ-PN . Thus, we get several characterizations of a λ-collectionwise normal andcountably paracompact space by combining one of the conditions in Theorems 5.1
and 5.2 with one of the conditions (1)–(9) in Theorem 3.1 and Remark 3.2. In
particular, we have the following consequence which is a mapping analogue of theSELECTIONS AND SANDWICH-LIKE PROPERTIES

43

Dowker’s characterization [6] of collectionwise normal and countably paracompact
spaces.
Corollary 5.3. For an infinite cardinal λ, the following conditions on a normal
space X are equivalent:(1) X is λ-collectionwise normal and countably paracompact.
(2) For every upper semi-continuous map g : X → c0 (λ), there exists a continuous map f : X → c0 (λ) such that g ≤ f .
For other characterizations of collectionwise normal countably paracompact spaces,see [21].
We complete this paper with the following characterization of paracompact
spaces which is just like Corollary 5.3, only it deals with maps to C0 (λ), where
λ is the space of all ordinals less than λ with the usual order topology.
Theorem 5.4. For an infinite cardinal λ, the following conditions on a normal
space X are equivalent:(1) X is λ-paracompact.
(2) For every space Y , with w(Y ) ≤ λ, and for every upper semi-continuous
map g : X → C0 (Y ), there exists a continuous map f : X → C0 (Y ) suchthat g ≤ f .
(3) For every upper semi-continuous map g : X → C0 (λ), there exists a continuous map f : X → C0 (λ) such that g ≤ f .
In the proof of Theorem 5.4, the normality of X is only used to apply Michael’s
result in the implication (1) ⇒ (2). Thus, we have the following corollary.
Corollary 5.5. The following conditions on a Hausdorff space X are equivalent:(1) X is paracompact.
(2) For every space Y and every upper semi-continuous map g : X → C0 (Y ),there exists a continuous map f : X → C0 (Y ) such that g ≤ f .
(3) For every infinite cardinal λ and every upper semi-continuous map g : X →C0 (λ), there exists a continuous map f : X → C0 (λ) such that g ≤ f .
Concerning the statements of Corolary 5.5, the following question naturally
arises.
Problem 5.6. Is a space X paracompact provided for every space Y and every
two maps g, h : X → C0 (Y ) such that g is upper semi-continuous, h is lower
semi-continuous and g ≤ h, there exists a continuous map f : X → C0 (Y ) withg ≤ f ≤ h?
References[1] R. A. Aló and H. L. Shapiro, Normal topological spaces, Cambridge University Press, London,
1974.
[2] M. Choban and V. Valov, On a theorem of E. Michael on selections, C. R. Acad. Bulgare
Sci. 28 (1975), 871–873, (in Russian).
[3] J. B. Conway, A course in Functional Analysis, GTM 96 (1985), Springer.
[4] J. Dieudonne, Une généralisation des espaces compacts, J. de Math. Pures et Appl. 23 (1944),
65–76.
[5] C. H.Dowker, On countably paracompact spaces, Canad. J. of Math. 3 (1951), 219–224.
[6] C. H. Dowker, Homotopy extension theorems, Proc. London Math. Soc. 6 (1956), 100-116[7] J. Dugundji, An extension of Tietze’s theorem, Pacific J. Math. 1 (1951), 353–367.

44

V. GUTEV, H. OHTA, AND K. YAMAZAKI[8] R. Engelking, General Topology, Revised and completed edition, Heldermann Verlag, Berlin,
1989.
[9] M. K. Fort, Jr., A unified theory of semi-continuity, Duke Math. J. 16 (1949), 237–246.
[10] L. Gillman and M. Jerison, Rings of Continuous Functions, Van Nostrand, New York, 1960.
[11] V. Gutev, Weak factorizations of continuous set-valued mappings, Topology Appl. 102(2000), 33–51.
[12] V. Gutev, Generic extensions of finite-valued u.s.c. selections, Topology Appl. 104 (2000),
101–118.
[13] M. Kandô, Characterization of topological spaces by some continuous functions, J. Math.
Soc. Japan 6 (1954), 45–54.
[14] M. Katětov, On real-valued functions in topological spaces, Fund. Math. 38 (1951), 85–91.
[15] M. Katětov, Correction to “On real-valued functions in topological spaces”, Fund. Math. 40(1953), 203–205.
[16] L. L. Krajewski, On expanding locally finite collections, Canad. J. Math. 23 (1971), 58–68.
[17] J. Mack, Directed covers and paracompact spaces, Canad. J. Math. 19 (1967), 649–654.
[18] E. Michael, Point-finite and locally finite coverings, Canad. J. Math. 7 (1955), 275–279.
[19] E. Michael, Continuous selections I, Ann. of Math. 63 (1956), 361–382.
[20] A. W. Miller, Special subsets of the real line, Handbook of Set-theoretic Topology (K. Kunen
and J. E. Vaughan, eds), North-Holland, Amsterdam, 1984, pp. 201–233.
[21] K. Miyazaki, Characterizations of paracompact-like, where D = {z ∈ C : |z| ≤ 1} properties
by means of set-valued semi-continuous selections, Proc. Amer. Math. Soc. 129 (2001), 2777–
2782.
[22] K. Morita, Products of normal spaces with metric spaces, Math. Ann. 154 (1964), 365–382.
[23] S. Nedev, Selection and factorization theorems for set-valued mappings, Serdica 6 (1980),
291–317.
[24] T. Przymusiński, Collectionwise normality and extensions of continuous functions, Fund.
Math. 98 (1978), 75–81.
[25] V. Šediva, On collectionwise and strongly paracompact spaces, Czech Math. J. 9 (1959),
50–62, (in Russian).
[26] I. Singer, Basis in Banach spaces I, Die Grundlehren der Math. Wissenschaften Band 154(1970), Springer.
[27] J. C. Smith and L. L. Krajewski, Expandability and collectionwise normality, Trans. Amer.
Math. Soc. 160 (1971), 437–451.
[28] F. D. Tall, Normality versus collectionwise normality, Handbook of Set-theoretic Topology(K. Kunen and J. E. Vaughan, eds), North-Holland, Amsterdam, 1984, pp. 685–732.
[29] H. Tong, Some characterizations of normal and perfectly normal spaces, Duke Math. J. 19(1952), 289–292.
[30] P. Zenor, Countable paracompactness of Fσ -sets, Proc. Amer. Math. Soc. 55 (1976), 201–202.
(V. Gutev) School of Mathematical and Statistical Sciences, Faculty of Science,
University of Natal, King George V Avenue, Durban 4041, South Africa
E-mail address: gutev@nu.ac.za(H. Ohta) Faculty of Education, Shizuoka University, Ohya, Shizuoka, 422-8529 Japan
E-mail address: echohta@ipc.shizuoka.ac.jp(K. Yamazaki) Institute of Mathematics, University of Tsukuba, Tsukuba-shi, Ibaraki,305-8571 Japan
E-mail address: kaori@math.tsukuba.ac.jp

Proceedings of General Topology Symposium, p. 45 - p. 53, 2002

http://gt2002.h.kobe-u.ac.jp/

THE BAIRE SPACE ORDERED BY EVENTUAL DOMINATION:
SPECTRAJÖRG BRENDLE∗
Abstract. These are notes of the author’s talk on various types of spectra associated naturally with the eventually domination ordering on the Baire spaceω ω , given at the General Topology Symposium at Kobe University in December
2002. The report comes in two parts: in the first half, we present an outlineof the lecture, giving ideas of some of the arguments without going too deeply
into details. The second part presents the technical niceties of some proofs.
This part was circulated previously under the title Chubu Marginalia [2].

1. Outline of the lecture
The Baire space ω ω is the set of all functions from the natural numbers ω to ω,
equipped with the product topology of the discrete topology. Given f, g ∈ ω ω saythat g eventually dominates f (f ≤∗ g in symbols) if f (n) ≤ g(n) holds for all but
finitely many n ∈ ω.
A family F ⊆ ω ω is called unbounded if there is no g ∈ ω ω with f ≤∗ g for allf ∈ F. F ⊆ ω ω is said to be dominating if for all g ∈ ω ω there is f ∈ F withg ≤∗ f . It is easy to see that a dominating family is also unbounded. We let b :=min{|F|; F ⊆ ω ω unbounded}, the (un)bounding number. d := min{|F|; F ⊆ ω ω
dominating} is the dominating number. The cardinal invariants b and d characterize
the combinatorial structure of (ω ω , ≤∗ ).
Fact 1.1. ℵ1 ≤ b ≤ cf (d) ≤ d ≤ c and b is regular.
(Here, cf means cofinality, and c = |2ω | = |R| stands for the size of the continuum.)
As a leitmotiv for this talk we address: What other notions can be used todescribe the combinatorial structure of (ω ω , ≤∗ )?
♣♣♣For a given preorder (P, ≤) (that is, ≤ is reflexive and transitive, but not necessarily
antisymmetric), Fuchino and Soukup [4] defined the following four spectra.
(i) the unbounded chain spectrum S↑ (P ), the set of all regular cardinals κsuch that there is an unbounded increasing chain of length κ in P ;(ii) the hereditarily unbounded set spectrum Sh (P ), the set of all cardinals κsuch that there is A ⊆ P of size κ such that all subsets of A of size κ are
unbounded in P while all subsets of A of size less than κ are bounded inP;∗

The author is supported by the Kobe Technical Club KTC (神戸大学工学振興会)
45

46

JÖRG BRENDLE(iii) the unbounded set spectrum S(P ), the set of all cardinals κ such that there
is unbounded A ⊆ P of size κ such that all subsets of A of size less thanκ are bounded in P ;(iv) the unbounded family spectrum Ss (P ), theSset of all cardinals κ such that
there is a family F ⊆ P(P ) ofSsize κ with F being unbounded while for
all G ⊆ F of size less than κ, G is bounded.
Clearly S↑ (P ) ⊆ Sh (P ) ⊆ S(P ) ⊆ Ss (P ).
♣♣♣We shall study the connection between these spectra for (P, ≤) = (ω ω , ≤∗ ). It iseasy to see that b = min S↑ (ω ω , ≤∗ ) = min Ss (ω ω , ≤∗ ). Fuchino and Soukup [4]asked:
Question 1.2. (Fuchino, Soukup) Is S↑ (ω ω , ≤∗ ) ( Sh (ω ω , ≤∗ ) consistent? Is
Sh (ω ω , ≤∗ ) ( S(ω ω , ≤∗ ) consistent? Is S(ω ω , ≤∗ ) ( Ss (ω ω , ≤∗ ) consistent?
More concretely:
Question 1.3. Is ℵ2 ∈ Sh (ω ω , ≤∗ ) \ S↑ (ω ω , ≤∗ ) consistent? Is ℵ2 ∈S(ω ω , ≤∗ ) \ Sh (ω ω , ≤∗ ) consistent? Is ℵ2 ∈ Ss (ω ω , ≤∗ ) \ S(ω ω , ≤∗ ) consistent?
Define b0 to be the supremum of cofinalities of unbounded well–ordered chainsin ω ω . b∗ is the minimal κ such that every unbounded family F ⊆ ω ω has an
unbounded subfamily of size κ.
Fact 1.4. b0 = sup(S↑ (ω ω , ≤∗ )), b∗ = sup(S(ω ω , ≤∗ )), as well as b ≤ b0 ≤ b∗ ≤ d.
The first instance of the above question has been answered a couple of years ago
in joint work with LaBerge [1].
Theorem 1.5. It is consistent that ℵ2 ∈ Sh (ω ω , ≤∗ ) \ S↑ (ω ω , ≤∗ ) and c = ℵ2 .
♣♣♣We proceed to present the main ideas of the proof of Theorem 1.5. Let A be a set.
CA denotes Cohen forcing with index set A, that is, the collection of finite partial
functions s : A × ω → 2 ordered by reverse inclusion. The latter means that t ≤ s
iff t ⊇ s. For each a ∈ A, CA adds a Cohen–generic real ca . Let X ⊆ A. Then
CX < ◦ CA (we say CX completely embeds into CA ), i.e., CX is a subforcing ofCA , and the “intermediate” generic extension via CX is a submodel of the genericextension via CA .
As usual, D denotes Hechler forcing, that is, the collection of all pairs (s, f )where s ∈ ω <ω , f ∈ ω ω and s ⊆ f . We order D by stipulating (t, g) ≤ (s, f ) ifft ⊇ s and g ≥ f everywhere. D generically adds a real d which eventually dominates
all ground model reals.
In the extension VX via CX , let DX denote Hechler forcing in the sense of VX .
This means of course that DX = {(s, f ); f ∈ VX ∩ ω ω }.
We are ready to describe the forcing we are going to use. Let V |= CH. Set
Y<ω
P = Cω2 ?
Ḋαα<ω2(Here, ? denotes iteration as usual, and the superscript <ω means we are forcing
with the finite support product.) Let W be the generic extension of V via P.
Let us list a few properties of the forcing P and of the resulting model W .

THE BAIRE SPACE ORDERED BY EVENTUAL DOMINATION: SPECTRA

47(a) P is ccc; so it preserves cardinals.
(b) As above, let cα , α < ω2 , denote the Cohen reals adjoined by Cω2 . Also
let dα , α < ω2 , denote the Hechler reals adjoined by Dα (over the genericextension via Cω2 ). Then cα ≤∗ dβ ⇐⇒ α < β.
(c) Let ġ be a P–name for a real. Then °P |{α < ω2 ; ċα ≤ ġ}| ≤ ℵ1 (this isso because of the way P factorizes as a product).
(d) In W , let F = {cα ; α < ω2 }. Then F witnesses ℵ2 ∈ Sh (ω ω , ≤∗ ), by (b)and (c).
(e) ℵ2 ∈/ S↑ (ω ω , ≤∗ ). In fact, there are no well–ordered ω2 –sequences inω(ω , ≤∗ ) in W .
We briefly describe the proof of (e) for this is the heart of the matter. It is basedon the following two lemmata.
Lemma 1.6. (Kunen) Assume CH. There are no well–ordered ω2 –sequences in(ω ω , ≤∗ ) in VA (the extension via CA ).
This is proved via a standard “isomorphism–of–names” trick.
Lemma 1.7. (Interpolation trick) Let Q and R be p.o.’s. Assume ḣ1 and ḣ2 are Q–
names for reals, and ġ is an R–name for a real such that (q, r) °Q×R ḣ1 ≤∗ ġ ≤∗ ḣ2 .
Then q °Q “∃f ∈ ω ω ∩ V : ḣ1 ≤∗ f ≤∗ ḣ2 .”Given a sequence {f˙α ; α < ω2 } of P–names use a ∆–system argument to makesupports nice (this uses CH) and step into an intermediate extension (essentially aCohen extension) such that the f˙α are adjoined by a product over this intermediate
extension. Assume °P “the f˙α are well–ordered by ≤∗ .” Using Lemma 1.7, we caninterpolate reals gα from the intermediate extension. This contradicts Lemma 1.6.
¤♣♣♣
Recently we proved:
Theorem 1.8. It is consistent that ℵ2 ∈ S(ω ω , ≤∗ ) \ Sh (ω ω , ≤∗ ) and c = ℵ3 .
With more work one can do better.
Theorem 1.9. It is consistent that ℵ2 ∈ S(ω ω , ≤∗ ) \ Sh (ω ω , ≤∗ ) and c = ℵ2 .
The proof of Theorem 1.8 was briefly sketched at the end of our lecture. Since
we include a detailed account of these results below in Section 2, we shall not givethis outline here.
We conjecture that similar techniques can be used to get a model where ℵ2 ∈Ss (ω ω , ≤∗ ) \ S(ω ω , ≤∗ ), but so far we have been unable to prove this.
More recently, we obtained:
Theorem 1.10. (see [3]) It is consistent that c = ℵ2 , ℵ2 ∈ S(ω ω , ≤∗ ) and there isno definable relation A ⊆ (ω ω )2 such that A well–orders some subset {fα ∈ ω ω ; α <ω2 } of the reals (i.e., α < β iff (fα , fβ ) ∈ A).
In this model, we obviously have ℵ2 ∈/ S↑ (ω ω , ≤∗ ). As a matter of fact, it is nothω∗difficult to show that ℵ2 ∈ S (ω , ≤ ) is sufficient to construct a definable well–ordering of a set of reals of length ω2 [3]. Therefore, Theorem 1.10 is a strengtheningof Theorem 1.9. See [3] for details.

48

JÖRG BRENDLE

2. Technical details
Assume A and B are disjoint sets and G ⊆ A × B is a bipartite graph. We willdefine the poset P = PG .
For X ⊆ A, CX denotes the algebra for adding |X| many Cohen reals with index
set X. Clearly, CX <◦ CY for X ⊆ Y . Fix b ∈ B and let Xb = {a ∈ A; (a, b) ∈ G}.
In the generic extension via CXb we have Hechler forcing D which we denote by
DXb . The p.o. P we force with is a two–step iteration given byY<ω
P = CA ?
ḊXb
Q<ω

b∈B

means that we take the finite support product of the partial orders DXb
Herein the generic extension via CA . Conditions p ∈ P can be canonically representedbyp = (spa , (tpb , ḣpb ); a ∈ F p , b ∈ Gp )where F p ⊆ A and Gp ⊆ B are finite. So this means spa ∈ C{a} and °CXb (tpb , ḣpb ) ∈ḊXb . We let supp(p) = F p ∪ Gp , the support of p.
For later use we notice two facts.
Q<ωFact 2.1. Given X ⊆ A and Y ⊆ B, we form PGX,Y = PX,Y = CX ? b∈Y ḊX∩Xb .
Then PX,Y <◦ P.
Proof. Let p = (spa , (tpb , ḣpb ); a ∈ F p , b ∈ Gp ) ∈ P. We need to find a reduction
p0 ∈ PX,Y of p. We shall have p0 = (spa0 , (tpb 0 , ḣpb 0 ); a ∈ F p0 , b ∈ Gp0 ) where• F p0 = F p ∩ X• Gp0 = Gp ∩ Y• spa0 = spa for a ∈ F p0
• tpb 0 = tpb for b ∈ Gp0
and the ḣpb 0 are given as follows.
Write sp = (spa ; a ∈ F p ) ∈ CA . So sp0 = (spa ; a ∈ F p0 ) = sp ¹CX ∈ CX . Forb ∈ Gp0 let supp(ḣpb ) ⊆ Xb be what is needed to decide ḣpb . By ccc–ness of CXb ,S
supp(ḣpb ) is countable. Set X 0 = X ∩ b∈Gp0 supp(ḣpb ). Also let X 1 = A \ X. So
X 0 and X 1 are disjoint, and X 0 is countable. Let {sn ; n ∈ ω} enumerate theconditions of CX 0 which are below sp0 ¹CX 0 (= sp ¹CX 0 ). Recursively find numbers`n,k,b (k ≤ n and b ∈ Gp0 ), and conditions tn ∈ CX 0 , un ∈ CX 1 such that tn ≤ sn
and the un form a decreasing chain below sp ¹CX 1 and such thattn ∪ un ° ḣpb (k) = `n,k,b for k ≤ n and b ∈ Gp0 .
This is clearly possible. Note that for b ∈ Gp0 , we actually have (tn ∪un )¹supp(ḣpb ) °ḣpb (k) = `n,k,b for k ≤ n.
Now define the CX∩Xb –name ḣpb 0 by stipulating that tn ¹supp(ḣpb ) forces ḣpb 0 (k) =`n,k,b for k ≤ n. We need to verify this is well–defined, that is, no two compatible
conditions force contradictory statements. To see this, fix k and assume tn ¹supp(ḣpb )and tm ¹supp(ḣpb ) are compatible for some m, n ≥ k. Since the uj are decreasing,(tn ∪ un )¹supp(ḣpb ) and (tm ∪ um )¹supp(ḣpb ) must be compatible as well. So theymust force the same value to ḣpb (k), i.e., `n,k,b = `m,k,b as required.
This completes the definition of p0 . We leave it to the reader to verify that given
q0 ≤ p0 , q0 ∈ PX,Y , there is a common extension q ∈ P of q0 and p.
¤

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49

This entails for example that if f˙ and ġ are PX,Y –names, then°P f˙ ≤∗ ġ ⇐⇒°PX,Y f˙ ≤∗ ġ

Fact 2.2. Given X0 , X1 ⊆ A, Y0 , Y1 ⊆ B, X = X0 ∩ X1 , Y = Y0 ∩ Y1 , andbijections v : X0 → X1 , w : Y0 → Y1 fixing X and Y respectively, if (a, b) ∈ G
iff (v(a), w(b)) ∈ G for all a ∈ X0 and all b ∈ Y0 , then PX0 ,Y0 and PX1 ,Y1 are
X0 ,Y0canonically isomorphic via an isomorphism φ = φX
fixing PX,Y .
1 ,Y1Note this means that if f˙ is a PX,Y –name for a real, ġ is a PX0 ,Y0 –name for areal and p ∈ PX0 ,Y0 , k ∈ ω are such that
p °X0 ,Y0 ∀n ≥ k (f˙(n) ≤ ġ(n))
then if φ(ġ) is the image of the name ġ we haveφ(p) °X1 ,Y1 ∀n ≥ k (f˙(n) ≤ φ(ġ)(n))φ(p)

Here, φ(p) is gotten from p as follows. F φ(p) = v(F p ), Gφ(p) = w(Gp ), sa
=
φ(p)
φ(p)
is the image of ḣpw−1 (b)spv−1 (a) for a ∈ F φ(p) , tb = tpw−1 (b) for b ∈ Gφ(p) , and ḣbunder the isomorphism between CX0 and CX1 induced by v, for all b ∈ Gφ(p) .
Furthermore p and φ(p) are compatible (in PX0 ∪X1 ,Y0 ∪Y1 ). (may not need this???
details see handwritten notes)
Denote the Cohen reals added by CA by {ca ; a ∈ A}, and the Hechler reals
adjoined by the finite support product, by {db ; b ∈ B}.
♣♣♣
Proof of Theorem 1.8. Assume GCH. Let A and B be sets of size ℵ3 , without
loss A = B = ω3 . Recursively construct a bipartite graph G ⊆ A × B such that(i) for all countable Y ⊆ B there is a ∈ ω2 such that (a, b) ∈/ G for all b ∈ Y(ii) for all X ⊆ A with X ⊆ ω2 and |X| ≤ ℵ1 , there is b ∈ B such that(a, b) ∈ G for all a ∈ X(iii) for all pairwise disjoint countable X, Xα ⊆ A (α < ω2 ), all pairwise
disjoint
S
countable Y, Yα ⊆ S
B (α < ω2 ), all {xn ; n ∈ ω} ⊆ 2X∪ α<ω2 Xα , all{yn ; n ∈ ω} ⊆ 2Y ∪ α<ω2 Yα , and all bipartite graphs G0 ⊆ ω × ω, thereare Ω ⊆ ω2 of size ℵ2 as well as {an ; n ∈ ω} ⊆ A disjoint from X, Xα and{bn ; n ∈ ω} ⊆ B disjoint from Y, Yα such thatfor all a ∈ X ∪for all b ∈ Y ∪S
S(a, bn ) ∈ G ⇐⇒ xn (a) = 1α∈Ω Xα and all n ∈ ω,

(an , b) ∈ G ⇐⇒ yn (b) = 1α∈Ω Yα and all n ∈ ω and(an , bm ) ∈ G ⇐⇒ (n, m) ∈ G0
for all n, m ∈ ω.
This is done by a recursive construction of length ω3 , taking care of (ii) in the firststep and of (iii) in the remaining steps. Since we are allowed to thin out to a set Ω
we can guarantee that (i) stays valid along the construction. For example we could
stipulate that for all b ∈ B, the set Xb = {a ∈ ω2 ; (a, b) ∈ G} is non–stationary inω2 . Details are left to the reader.
It is clear by |A| = |B| = ℵ3 that the size of the continuum will be ℵ3 .

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Lemma 2.3. The Cohen reals {ca ; a < ω2 } are unbounded in the P–extension, yetevery subfamily of size ℵ1 is bounded.
Proof. Let ġ be a P–name for a real. There are X ⊆ A countable
and Y ⊆ B
S
countable such that ġ is a PX,Y –name. By (i) there is a ∈ ω2 \ b∈Y Xb . Note thatfor such a, PX∪{a},Y = PX,Y × C{a} . That is, ca is still Cohen over the extension
via PX,Y and, in particular, it is not bounded by the interpretation of ġ.
That every subfamily of size ℵ1 is bounded immediately follows from property(ii).
¤
Main Lemma 2.4. In the P–extension, the following holds: Assume {fα ; α < ω2 }is such that for all β < ω2 , {fα ; α < β} is bounded. Then there is Ω ⊆ ω2 , |Ω| = ℵ2 ,such that {fα ; α ∈ Ω} is bounded.
Proof. Let {f˙α ; α < ω2 } and {ġβ ; β < ω2 } be P–names such that°P f˙α ≤∗ ġβ
for all α < β < ω2 . By CH and a ∆–system argument we may assume that there
are pairwise disjoint countable sets X, Xα , Aβ ⊆ A (α, β < ω2 ) and pairwise disjoint
countable sets Y, Yα , Bβ ⊆ B (α, β < ω2 ) such that all f˙α are PX∪Xα ,Y ∪Yα –names
and all ġβ are PX∪Aβ ,Y ∪Bβ –names. By the remark after Fact 2.1,
°PX∪Xα ∪Aβ ,Y ∪Yα ∪Bβ f˙α ≤∗ ġβ
for all α < β < ω2 .
We may further assume that for α 6= β we have bijections vα,β : X ∪Aα → X ∪Aβand wα,β : Y ∪Bα → Y ∪Bβ fixing X and Y respectively such that for all a ∈ X ∪Aαand all b ∈ Y ∪ Bα , we have (a, b) ∈ G iff (vα,β (a), wα,β (b)) ∈ G. By Fact 2.2, thisX∪Aα ,Y ∪Bαmeans that we get an isomorphism φ = φαβ = φX∪Aβ ,Y ∪Bβ between PX∪Aα ,Y ∪Bα
and PX∪Aβ ,Y ∪Bβ . We may also suppose that this induced isomorphism identifies
the corresponding names ġα and ġβ , i.e., φ(ġα ) = ġβ . List Aα = {aα,n ; n ∈ ω}such that vα,β (aα,n ) = aβ,n and Bα = {bα,n ; n ∈ ω} such that wα,β (bα,n ) = bβ,n .
Let G0 be such that (n, m) ∈ G0 iff (aα,n , bα,m ) ∈ G (note this is independent of the
choice of α).
Next, for each α there is a cofinal set Cα ⊆ ω2 \(α+1) such that for all β, β 0 ∈ Cα :(a, b) ∈ G iff (a, wβ,β 0 (b)) ∈ G for all a ∈ Xα and all b ∈ BβS, and (a, b) ∈ G iff(vβ,β 0 (a), b) ∈ G for all a ∈ Aβ and all b ∈ Yα . Define xn : X ∪ α<ω2 Xα → 2 by
 1 if (a ∈ X and (a, bβ,n ) ∈ G) or(a ∈ Xα and (a, bβ,n ) ∈ G for β ∈ Cα )xn (a) =
0 otherwiseSand yn : Y ∪ α<ω2 Yα → 2 by
 1 if (b ∈ Y and (aβ,n , b) ∈ G) or(b ∈ Yα and (aβ,n , b) ∈ G for β ∈ Cα )yn (b) =
0 otherwise(again this is independent of the choice of β).
We may now find Ω ⊆ ω2 of size ℵ2 and Aω2 = {an ; n ∈ ω} ⊆ A, Bω2 = {bn ; n ∈ω} ⊆ B satisfying the requirements of clause (iii). Note this means that for any
α ∈ Ω and any β ∈ Cα we have functions vβ,ω2 : X ∪ Xα ∪ Aβ → X ∪ Xα ∪ Aω2

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51

and wβ,ω2 : Y ∪ Yα ∪ Bβ → Y ∪ Yα ∪ Bω2 fixing X ∪ Xα and Y ∪ Yα respectively
such that(a, b) ∈ G ⇐⇒ (vβ,ω2 (a), wβ,ω2 (b)) ∈ G
for all a ∈ X ∪ Xα ∪ Aβ and all b ∈ Y ∪ Yα ∪ Bβ . By Fact 2.2 we get a canonicalisomorphism φβω2 : PX∪Xα ∪Aβ ,Y ∪Yα ∪Bβ → PX∪Xα ∪Aω2 ,Y ∪Yα ∪Bω2 . Let ġω2 be theφβω2 –image of ġβ (note this is independent of the choice of β). By the discussion afterFact 2.2 and the assumption on f˙α and ġβ , we see that °PX∪Xα ∪Aω ,Y ∪Yα ∪Bω f˙α ≤∗22ġω2 . By the remark after Fact 2.1, this is also true in the P–generic extension. ¤¤♣♣♣Proof of Theorem 1.9. Assume CH. Construct first a bipartite graph H ⊆ ω1 × ω1
such that(i) for all countable Y ⊆ ω1 there is x ∈ ω1 such that (x, y) ∈/ H for all y ∈ Y(ii) for all countable X = {xn ; n ∈ ω} ⊆ ω1 , all countable Y ⊆ ω1 and all
f : ω × X × ω → 2 there are countable X 0 = {x0n,m ; n, m ∈ ω} ⊆ ω1
0
; m ∈ ω} ⊆ ω1 disjoint from Y
disjoint from X and countable Y 0 = {ym0such that (xn , ym ) ∈ H for all n, m ∈ ω,
(x0n,m , y) ∈ H ⇐⇒ (xn , y) ∈ H
for all n, m ∈ ω and y ∈ Y , as well as(x0n,m , y`0 ) ∈ H ⇐⇒ f (m, xn , `) = 1
for all n, m, ` ∈ ω.
Using CH, H can be constructed by an easy recursion of length ω1 , taking care of(i) and (ii) alternately.
˙ 1 . G ⊆ A × B is the bipartite graph given
Next let A = ω2 × ω1 and B = ω2 ∪ω
by((ζ, η), ζ 0 ) ∈ G((ζ, η), η 0 ) ∈ G⇐⇒⇐⇒ζ < ζ0(η, η 0 ) ∈ H

for all ζ, ζ 0 ∈ ω2 and η, η 0 ∈ ω1 . We consider forcing with P = PG as before. Since|A| = |B| = ℵ2 , it is immediate that the size of the continuum will be ℵ2 .
By (i) and definition of G, properties analogous to (i) and (ii) of the previous
proof hold and it follows that the Cohen reals witness that ℵ2 ∈ S(ω ω , ≤∗ ). So weare left with showing:
Main Lemma 2.5. In the P–extension, the following holds: Assume {fα ; α < ω2 }is such that for all β < ω2 , {fα ; α < β} is bounded. Then there is Ω ⊆ ω2 , |Ω| = ℵ2 ,such that {fα ; α ∈ Ω} is bounded.
Proof. As in the previous proof we have f˙α , ġβ and countable pairwise disjoint
sets X, Xα , Aβ ⊆ A as well as countable pairwise disjoint sets Y, Yα , Bβ ⊆ B,α, β < ω2 . Let Y 0 = Y ∩ ω2 and let Y 1 = Y ∩ ω1 , the two disjoint pieces Y is
made off. By enlarging supports, if necessary, we may further assume there areθ < ω1 and ζα , ξα , α < ω2 , such that X = X 0 × θ, Xα = Xα0 × θ, Aβ = A0β × θ,
Y 1 ⊆ θ, Yα , Bβ ⊆ ω2 , and sup(X 0 ) < ζ0 < ... < ζα < min(Xα0 ) < sup(Xα0 ) <ξα < min(A0α ) < sup(A0α ) < ζα+1 ... and sup(Y 0 ) < ζ0 < ... < ζα < min(Yα ) <sup(Yα ) < ξα < min(Bα ) < sup(Bα ) < ζα+1 ... This means that for all α < β, all

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JÖRG BRENDLE

a ∈ X ∪ Xα and all b ∈ Bβ we have (a, b) ∈ G and for all a ∈ Aβ and all b ∈ Y 0 ∪ Yα
we have (a, b) ∈/ G. List θ = {xn ; n ∈ ω}. Also list Aβ = {(aβ,m , xn ); n, m ∈ ω}
(so A0β = {aβ,m ; m ∈ ω}) and Bβ = {bβ,m ; m ∈ ω} in such that way that((aα,m , xn ), bα,` ) ∈ G iff ((aβ,m , xn ), bβ,` ) ∈ G, for all α, β and all n, m, ` (note thatby definition of G, this depends only on m and ` and not on n; this is irrelevant,however).
Apply (ii) with the X there being {xn ; n ∈ ω} = θ, the Y there being Y 1 ⊆ θ andf given by f (m, xn , `) = 1 iff ((aβ,m , xn ), bβ,` ) ∈ G. We get X 0 = {x0n,m ; n, m ∈ ω}0and Bω2 = Y 0 = {ym; m ∈ ω} as stipulated in (ii). Choose any a0 > ζ0 from ω2 .
Let Aω2 = {a0 } × X 0 . For α < β, define v = vβ,ω2 : X ∪ Xα ∪ Aβ → X ∪ Xα ∪ Aω20
and w = wβ,ω2 : Y ∪ Bβ → Y ∪ Bω2 by v(aβ,m , xn ) = (a, x0n,m ) and w(bβ,m ) = ym
(identity otherwise). Then we have (a, b) ∈ G iff (v(a), w(b)) ∈ G for all a ∈X ∪ Xα ∪ Aβ and all b ∈ Y ∪ Bβ and all α < β. (check details!) By Fact 2.2, thismeans we get an isomorphism φ = φβω2 : PX∪Xα ∪Aβ ,Y ∪Bβ → PX∪Xα ∪Aω2 ,Y ∪Bω2 . As
in the previous proof, let ġω2 be the φβω2 –image of ġβ . The following claim finishes
the proof.
Main Claim 2.6. Given p ∈ P and α0 < ω2 , there are q ≤ p and α ≥ α0 with
p ° ġω2 ≥∗ f˙α .
Proof. Let p ∈ P and α0 < ω2 . Fix α ≥ α0 such that supp(p) ∩ Yα = ∅. We need tofind q ≤ p and k ∈ ω such that q ° “ġω2 (n) ≥ f˙α (n) for all n ≥ k.”First note we may assume p ∈ PX∪Xα ∪Aω2 ,Y ∪Bω2 . The point is that under thisassumption we will find q ∈ PX∪Xα ∪Aω2 ,Y ∪Yα ∪Bω2 . Now, if p is arbitrary, we may
first consider its reduction p̄ to PX∪Xα ∪Aω2 ,Y ∪Yα ∪Bω2 which actually must belong to
PX∪Xα ∪Aω2 ,Y ∪Bω2 . Thus we get q̄ ≤ p̄ as required with q̄ ∈ PX∪Xα ∪Aω2 ,Y ∪Yα ∪Bω2 .
So q̄ and p have a common extension.
Let p∗ = φ−1 (p) ∈ PX∪Xα ∪Aβ ,Y ∪Bβ where β > α is arbitrary. There are q ∗ ≤ p∗ ,q ∗ ∈ PX∪Xα ∪Aβ ,Y ∪Yα ∪Bβ , and k ∈ ω such that q ∗ ° “ġβ (n) ≥ f˙α (n) for all n ≥ k.”
The main technical difficulty with this proof now is that PX∪Xα ∪Aβ ,Y ∪Yα ∪Bβneed not be isomorphic to PX∪Xα ∪Aω2 ,Y ∪Yα ∪Bω2 , so we cannot go back directly.
Instead, we shall interpolate a name ḣα between f˙α and ġβ in such a way that ḣαdoes not depend on Yα nor on β.
The crux of the matter is that PX∪Xα ∪Aβ ,Y ∪Yα ∪Bβ is the amalgamation of
PX∪Xα ,Y ∪Yα and PX∪Xα ∪Aβ ,Y ∪Bβ over PX∪Xα ,Y . This is so because (a, b) ∈/ G
for all a ∈ Aβ and all b ∈ Yα which means that the Hechler reals adjoined by Yα do
not depend on the Cohen reals adjoined by Aβ .
To be more explicit, let q̄ ∗ be the projection of q ∗ to PX∪Xα ∪Aβ ,Y ∪Bβ . Next,
let q̄0∗ be the projection of q̄ ∗ to r.o.(PX∪Xα ,Y ) (note that, since we are working with projections in cBa’s here, q̄ ∗ does not necessarily belong to PX∪Xα ,Y ,but this is irrelevant here). Finally, define q0∗ ∈ r.o.(PX∪Xα ,Y ∪Yα ) as follows.
∗q0∗ ¹r.o.(PX∪Xα ,Y ) = q̄0∗ and q0∗ ¹Yα = q ∗ ¹Yα . The latter means that Gq0 ∩ Yα =
∗
∗
∗
∗
q∗
q∗
Gq ∩ Yα and (tb0 , ḣb0 ) = (tqb , ḣqb ) for b ∈ Gq0 ∩ Yα . This makes sense because all∗such ḣqb are C(X∪Xα ∪Aβ )∩Xb –names and therefore, as Aβ ∩ Xb = ∅, C(X∪Xα )∩Xb –names. Finally notice that q0∗ is the projection of q ∗ to r.o.(PX∪Xα ,Y ∪Yα ).
Step for a moment into the generic extension W via PX∪Xα ,Y such that q̄0∗ belongs to the generic filter. In W , forcing with PX∪Xα ∪Aβ ,Y ∪Yα ∪Bβ is nothing but
forcing with the product of PX∪Xα ,Y ∪Yα and PX∪Xα ∪Aβ ,Y ∪Bβ . (So PX∪Xα ∪Aβ ,Y ∪Yα ∪Bβis indeed the amalgamation as claimed above.) This means we may think of q ∗ asTHE BAIRE SPACE ORDERED BY EVENTUAL DOMINATION: SPECTRA

53

the pair (q0∗ , q̄ ∗ ). Since(q0∗ , q̄ ∗ ) ° “ġβ (n) ≥ f˙α (n) for all n ≥ k, ”
since f˙α is a PX∪Xα ,Y ∪Yα –name, and since ġβ is a PX∪Xα ∪Aβ ,Y ∪Bβ –name, we mayfind hα ∈ W ∩ ω ω such thatq ∗ ° “hα (n) ≥ f˙α (n) for all n ≥ k”
0andq̄ ∗ ° “ġβ (n) ≥ hα (n) for all n ≥ k.”(Simply let hα (n) = sup{m; there is r0∗ ≤ q0∗ in PX∪Xα ,Y ∪Yα such that r0∗ ° f˙α (n) =m}.)Back in the ground model, let ḣα be a PX∪Xα ,Y –name for hα . Let q̄ = φ(q̄ ∗ ).
Since φ fixes PX∪Xα ,Y , q̄ ≤ q̄0∗ and φ(ḣα ) = ḣα . So we get (see the discussion afterFact 2.2)q̄ ° “ġω2 (n) ≥ ḣα (n) for all n ≥ k.”
In PX∪Xα ∪Aω2 ,Y ∪Yα ∪Bω2 , define q such that q¹PX∪Xα ∪Aω2 ,Y ∪Bω2 = q̄ and q¹Yα =∗q0∗ ¹Yα (= q ∗ ¹Yα ). As before, the latter means that Gq ∩Yα = Gq0 ∩Yα and (tqb , ḣqb ) =
∗
∗
q
q
q∗
(tb0 , ḣb0 ) for b ∈ Gq ∩ Yα . (This is unproblematic because the ḣb0 are C(X∪Xα )∩Xb –names.) q ≤ q̄ is trivial but, using q̄ ≤ q̄0∗ and the way q0∗ and q were defined, we
also get q ≤ q0∗ . Therefore
q ° “ḣα (n) ≥ f˙α (n) for all n ≥ k”
so thatq ° “ġω2 (n) ≥ f˙α (n) for all n ≥ k, ”

as required.

¤¤¤
References[1] J. Brendle and T. LaBerge, Forcing tightness in products of fans, Fund. Math. 150 (1996)
211-226.
[2] J. Brendle, Chubu Marginalia, notes circulated in December 2002.
[3] J. Brendle and S. Fuchino, Coloring ordinals by reals, in preparation.
[4] S. Fuchino and L. Soukup, Chubu-2001 Notes, circulated notes.
The Graduate School of Science and Technology, Kobe University, Rokko–dai 1–1,
Nada–ku Kobe 657–8501, Japan
E-mail address: brendle@kurt.scitec.kobe-u.ac.jp

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Proceedings of General Topology Symposium, p. 55 - p. 57, 2002

http://gt2002.h.kobe-u.ac.jp/

PRODUCTS OF 2 SPACES AND PRODUCTS OF 3 SPACES
YASUSHI HIRATA AND NOBUYUKI KEMOTO

A space is paracompact (subparacompact, metacompact) if for every open coverU , there is a locally finite open (σ -locally finite closed, point finite open) refinement
V of U . A space is submetacompact if for every open cover U , there is a sequence{Vn : n ∈ ω} of open refinements of U such that for each point x, there is n ∈ ωsuch that Vn is point finite at x. It is well known that paracompactness implies
both of subparacompactness and metacompactness, and submetacompactness is
a common weakening of subparacompactness and metacompactness. We discusson these four properties (and its restricted version) of product spaces of ordinal
numbers with the usual order topology. The Greek letters α , β , γ ... denoteordinals. One of basic results is:
Proposition 1 ([8]). Let A and B be subspaces of α and X = A × B . Then
paracompactness, subparacompactness, metacompactness and submetacompactnessof X are equivalent.
Thus, these properties are equivalent for “products of subspaces”. But in general,these properties need not be equivalent for “subspaces of products”.
Proposition 2 ([7]).
(1) For every subspace X of α2 , metacompactness and submetacompactness of
X are equivalent. Thus paracompactness of X implies subparacompactness
and subparacompactness of X implies metacompactness.
(2) There is a metacompact subspace of (ω2 + 1)2 which is not subparacompact.
(3) There is a subparacompact subspace of (ω1 + 1)2 which is not paracompact.
Now let’s consider the restricted versions of these four properties. A space iscountably paracompact if for every countable open cover U , there is a locallyfinite open refinement V of U . Countable subparacomactness, countable metacompactness and countable submetacompactness are similarly defined. It is wellknown that countable metacompactness coincides with countable submetacompactness and it is a common weakening of countable paracompactness and countablesubparacompactness. But, as is witnessed by the product ω1 × (ω1 + 1) , countableparacompactness does not imply countable subparacompactness. The square of Sorgenfrey line is countably subparacompact but not countably paracompact. Thus
countable subparacompactness and countable paracompactness are incomparable.
Normal (Subnormal) countably metacompact spaces are countably paracompact(countably subparacompact), where a space is normal (subnormal) if every pair ofdisjoint closed sets are separated by disjoint open sets (Gδ -sets). Of course, normal
spaces are subnormal and countably subparacompact spaces are subnormal. But
it is well known that countable paracompactness and normality are incomparable.
Now we focus on these restricted properties of subspaces of ω1n , n ≤ ω .
55

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Y. HIRATA, AND N. KEMOTO

One of basic results about countable paracompactness is:
Proposition 3 ([3]). Let A and B be subspaces of ω1 and X = A × B . Then X
is countably paracompact iff X is normal iff A or B is non-stationary or A ∩ B
is stationary.
Note that for disjoint stationary sets A and B of ω1 , X = A × B is neither
normal nor countably paracompact. It is natural to ask whether normality andcoutable paracompactness are equivalent for every subspace of ω12 . A consistently
affirmative answer is known:
Proposition 4 ([6]). Assuming V = L or P M EA , normality and countable paracompactness are equivalent for every subspace of ω12 .
But, it still remains open whether this is a theorem of ZFC. Note that Proposition
5 bellow shows that normality implies countable paracompactness in ZFC.
For countable metacompactness, we have the following unexpected results.
Proposition 5 ([4]). All subspaces of ω12 are countably metacompact. Thus normal
subspaces of ω12 are countably paracompact.
Proposition 6 ([5]). All subspaces of ω1n , n ∈ ω , are countably metacompact, butthere is a subspace of ω1ω which is not countably metacompact.
For countable subparacompactness, we also have the following unexpected result.
Proposition 7 ([2]). All subspaces of ω12 are countably subparacompact.
So, as in the countably metacompact case, we have no doubt to conjecture thatall subspaces of ω1n , n ∈ ω , are countably subparacompact. But unfortunately weobtain:Theorem 8 ([1]). There is a subspace of ω13 which is not countably subparacompact.
The subspace X = {hα, β, γi ∈ ω13 : α ≤ β ≤ γ} \ {hα, α, αi ∈ ω13 : α < ω1 }of ω13 is the desired one. Indeed, we can show that the disjoint closed subsets
F0 = {hα, β, γi ∈ X : α = β} and F1 = {hα, β, γi ∈ X : β = γ} cannot be
separated by disjoint Gδ -sets. So X is not subnormal, and therefore X is a noncountably subparacompact subspace of {hα, β, γi ∈ ω13 : α ≤ β ≤ γ} . But somespecial subspaces are hereditarily countably subparacompact.
Theorem 9 ([1]). All subspaces of {hα, β, γi ∈ ω13 : α < β < γ} are countably
subparacompact. More generally, all subspaces of {x ∈ ω1n : x(0) < x(1) < ... <x(n − 1)} are countably subparacompact.
References[1] Y. Hirata and N. Kemoto, Separating by Gδ -sets in finite powers of ω1 , Fund. Math., toappear.
[2] N. Kemoto, Subnormality in ω12 , Top. Appl., 122 (2002), 287-296.
[3] N. Kemoto, H. Ohta and K. Tamano, Products of spaces of ordinal numbers, Top. Appl.,
45(1992), 245–260.
[4] N. Kemoto and K. D. Smith, The product of two ordinals is hereditarily countably metacompact, Top. Appl., 74 (1996), 91-96.
[5] N. Kemoto and K. D. Smith, Hereditarily countable metacompactness in finite and infinite
product spaces of ordinals, Top. Appl., 77 (1997), 57-63.

PRODUCTS OF 2 SPACES AND PRODUCT OF 3 SPACES

57[6] N. Kemoto, K. D. Smith and P. J. Szeptycki, Countable paracompactness versus normality
in ω12 , Top. Appl., 104 (2000), 141–154.
[7] N. Kemoto, K. Tamano and Y. Yajima, Generalized paracompactness of subspaces in products of two ordinals, Top. Appl., 104 (2000), 155–168.
[8] N. Kemoto and Y. Yajima, Orthocompactness in products, Tsukuba J. Math., 16 (1992),
407-422.
(Y. Hirata) Graduate School of Mathematics, University of Tsukuba, Ibaraki 3058571, Japan
E-mail address, Y. Hirata: yhira@jb3.so-net.ne.jp(N. Kemoto) Department of Mathematics, Faculty of Education, Oita University,
Dannoharu, Oita, 870-1192, Japan
E-mail address, N. Kemoto: nkemoto@cc.oita-u.ac.jp

Proceedings of General Topology Symposium, p. 59 - p. 66, 2002

http://gt2002.h.kobe-u.ac.jp/

HYPERSPACES WITH THE HAUSDORFF METRIC
MASAYUKI KURIHARA, KATSURO SAKAI, AND MASATO YAGUCHI

1. Introduction
This note is a presentation of the results obtained in the paper [8].
Let X = (X, d) be a metric space. The set of all non-empty closed sets in X isdenoted by Cld(X). On the subset Bdd(X) ⊂ Cld(X) consisting of bounded closed
sets in X, we can define the Hausdorff metric dH as follows:½¾
dH (A, B) = max sup d(x, A), sup d(x, B) ,
x∈B

x∈A

where d(x, A) = inf a∈A d(x, a). We denote the metric space (Bdd(X), dH ) by
BddH (X). On the whole set Cld(X), we allow dH (A, B) = ∞, but dH induces
the topology of Cld(X) like a metric does. The space Cld(X) with this topology isdenoted by CldH (X). When X is bounded, CldH (X) = BddH (X). Even though
X is unbounded, CldH (X) is metrizable. Indeed, let d¯ be the metric on X defined¯ y) = min{1, d(x, y)}. Then, d¯H is an admissible metric of CldH (X). It
by d(x,should be noted that each component of CldH (X) is contained in Bdd(X) or in
the complement Cld(X) \ Bdd(X). Thus, BddH (X) is a union of components of
CldH (X). On each component of CldH (X), dH is a metric even if it is containedin Cld(X) \ Bdd(X). Then, we regard every component of CldH (X) as a metricspace with dH .
When X is compact, it is well-known that CldH (X) (= BddH (X)) is an ANR(an AR)1 if and only if X is locally connected (connected and locally connected)[14]. However, in case X is non-compact, this does not hold. In [8], we construct a
metric AR X such that CldH (X) is not an ANR and give a condition on X suchthat CldH (X) is an ANR. Due to our result, CldH (X) can be an ANR even if X
is not locally connected.
2. Main results and a counter-example
Let X = (X, d) be a metric space. For A ⊂ X and γ > 0, we denote
N (A, γ) = {x ∈ X | d(x, A) < γ} and
N (A, γ) = {x ∈ X | d(x, A) 6 γ}.
When A = {a}, we write N ({a}, γ) = B(a, γ) and N ({a}, γ) = B(a, γ)
In [10], Michael introduced uniform AR’s and uniform ANR’s. A uniform ANR
is a metric space X with the property: for an arbitrary metric space Z = (Z, d)containing X isometrically as a closed subset, there exist a uniform neighborhood
1An ANR (an AR) means an absolute neighborhood retract (an absolute retract) for metrizable
spaces.
59

60

M. KURIHARA, K. SAKAI, AND M. YAGUCHIU of X in Z (i.e., U = N (X, γ) for some γ > 0) and a retraction r : U → X which is
uniformly continuous at X, that is, for each ε > 0, there is some δ > 0 such that if
x ∈ X, z ∈ U and d(x, z) < δ then d(x, r(z)) < ε. When U = Z in the above, X iscalled a uniform AR. A uniform ANR is a uniform AR if it is homotopically trivial,that is, all the homotopy groups are trivial. In [11], it is shown that a metric space
X is a uniform ANR if and only if every metric space Z containing X isometricallyas a dense subset is a uniform ANR and X is homotopy dense in Z, that is, there
exists a homotopy h : Z × I → Z such that h0 = idZ and ht (Z) ⊂ X for t > 0.
For each η > 0, a finite sequence (xi )ki=0 of points in X is called an η-chain ifd(xi , xi−1 ) < η for each i = 1, . . . , k, where k is the length of (xi )ki=0 . The diameterof (xi )ki=0 means diam{xi | i = 0, 1, . . . , k}. When x0 = x and xk = y, we call(xi )ki=0 an η-chain from x to y and we say that x and y are connected by (xi )ki=0 .
It is said that X is C-connected (connected in the sense of Cantor ) if each pair ofpoints in X are connected by an η-chain in X for any η > 0. Now, we say that
X is uniformly locally C ∗ -connected if for each ε > 0 there exists δ > 0 with thefollowing property:(ulC ∗ ) For each η > 0, there is k ∈ N such that each pair of δ-close points of X
are connected by an η-chain with the length 6 k and the diameter < ε.
It is easy to see that a metric space is uniformly locally C ∗ -connected if it is uniformly homeomorphic to a uniformly locally C ∗ -connected metric space.
A collection A of subsets of X is said to be uniformly discrete if there exists
some δ > 0 such that the δ-neighborhood B(x, δ) of each x ∈ X meets at most one
member of A, that is,
inf{dist(A, A0 ) | A 6= A0 ∈ A} > 0,
where dist(A, A0 ) = inf{d(x, x0 ) | x ∈ A, x0 ∈ A0 }. The following is the main resultin [8]:
Theorem. For every uniformly locally C ∗ -connected metric space X, the collectionof all components of CldH (X) is uniformly discrete and each component of the space
CldH (X) is a uniform AR, hence CldH (X) is an ANR and BddH (X) is a uniform
ANR.
By the main Theorem above, for every dense subset X of a convex set in anormed linear space, each component of CldH (X) is a uniform AR and BddH (X)
is a uniform ANR. Recently, Constantini and Kubiś [4] showed that BddH (X) isan AR if X is almost convex, where X is almost convex if for each x, y ∈ X andfor each s, t > 0 such that d(x, y) < s + t, there exists z ∈ X with d(x, z) < sand d(y, z) < t. This result follows from the main Theorem because every almost
convex metric space is uniformly locally C ∗ -connected.
Here, we construct a metric AR X such that CldH (X) is not an ANR.
S
Example 1. As a subspace of Euclidean space R3 , let X = n∈N∪{0} Xn , where
X0 = {(x, xz, z) ∈ R3 | x > 1, z ∈ I} and
Xn = {(x, y, 1/n) ∈ R3 | x > 1, 0 6 y 6 x/n} for n ∈ N.
Then, X is an AR. In fact, X is homeomorphic to (≈) the following space:µ¶[
I × {0} ∪{1/n} × [0, 1/n] × [1, ∞).
n∈N

HYPERSPACES WITH THE HAUSDORFF METRIC

61

Moreover, it can be proved that CldH (X) is not locally path-connected at A =N × {0} × {0}, hence is not an ANR.
In the above example, X is not a uniform ANR. The following is unknown:
Problem 1. For every uniform ANR X, is CldH (X) an ANR?
3. Lawson semilattices which are uniform ANR’s
A topological semilattice is a topological space S equipped with a continuous
operator ∨ : S × S → S which is reflexive, commutative and associative (i.e.,x ∨ x = x, x ∨ y = y ∨ x, (x ∨ y) ∨ z = x ∨ (y ∨ z)). A topological semilattice S iscalled a Lawson semilattice if S admits an open basis consisting of subsemilattices[9]. It is known that a metrizable Lawson semilattice is k-aspherical for each k > 0([4, Proposition 2.3]).
In [1], it is shown that a metrizable Lawson semilattice is an ANR (resp. anAR) if and only if it is locally path-connected (resp. connected and locally pathconnected). Here, we consider the condition that a metric Lawson semilattice is a
uniform ANR.
It is said that a metric space X is uniformly locally contractible if for each ε > 0,
there exist δ > 0 such that the δ-ball B(x, δ) at each x ∈ X is contractible inthe ε-ball B(x, ε). Every uniform ANR is uniformly locally contractible by [10,Proposition 1.5 and Theorem 1.6]. And, as is easily observed, every uniformly
locally contractible metric space is uniformly locally k-connected for all k > 0.
We have the following characterization:
Theorem 3.1. Let L = (L, d, ∨) be a metric Lawson semilattice such thatd(x ∨ x0 , y ∨ y 0 ) 6 max{d(x, y), d(x0 , y 0 )} for each x, x0 , y, y 0 ∈ L.
Then, the following are equivalent:(a) the collection of all components of L is uniformly discrete in L and eachcomponent of L is a uniform AR;(b) L is a uniform ANR;(c) L is uniformly locally contractible;(d) L is uniformly locally path-connected.
4. The uniform local C ∗ -connectedness
For two metric spaces X = (X, dX ) and Y = (Y, dY ), let C(X, Y ) be the collection of all continuous functions from X to Y . It is said that F ⊂ C(X, Y ) is uniformly equi-continuous if for each ε > 0, there is δ > 0 such that dY (f (x), f (x0 )) < ε
for each f ∈ F and x, x0 ∈ X with dX (x, x0 ) < δ. We can characterize the uniform
locall C ∗ -connectedness as follows:
Theorem 4.1. Let D be a countable dense subset of the unit interval I with theusual metric and 0, 1 ∈ D. Then, a metric space X = (X, d) is uniformly locallyC ∗ -connected if and only if for each ε > 0, there exist δ > 0 and F ⊂ C(D, X)
satisfying the following:(i) F is uniformly equi-continuous,(ii) diam f (D) < ε for every f ∈ F,(iii) for each δ-close x, y ∈ X, there is f ∈ F with f (0) = x and f (1) = y.

62

M. KURIHARA, K. SAKAI, AND M. YAGUCHIFor a complete metric space X and a dense subset D ⊂ I, since every uniformly
continuous map f : D → X extends over I, the following follows from Theorem 4.1:
Corollary 4.2. Every uniformly locally C ∗ -connected complete metric space is uniformly locally path-connected.
It is known that CldH (X) = (CldH (X), ∪) is a Lawson semilattice satisfying thefollowing condition:dH (A ∪ A0 , B ∪ B 0 ) 6 max{dH (A, B), dH (A0 , B 0 )}for each A, A0 , B, B 0 ∈ CldH (X).
Refer to [4, Proposition 2.4] (cf. the proof of [1, Fact 4]). The following can beproved:Theorem 4.3. For every uniformly locally C ∗ -connected metric space X, the space
CldH (X) is uniformly locally path-connected.
Combining this result with Theorem 3.4, we can obtain the main Theorem.
5. The uniformly local almost convexity
A metric space X = (X, d) is locally almost convex if each x ∈ X has a neighborhood U such that(lac) for each y, z ∈ U and for each s, t > 0 with s + t > d(y, z), there is w ∈ Xsuch that d(y, w) < s and d(w, z) < t.
It is said that X is uniformly locally almost convex if there is some δ > 0 such that(ulac) for each x, y ∈ X with d(x, y) < δ and for each s, t > 0 with s+t > d(x, y),there is some z ∈ X such that d(x, z) < s and d(y, z) < t.
Note that an almost convex metric space is uniformly locally almost convex. Obviously, a uniformly locally almost convex metric space is locally almost convex, but
the converse does not hold as the example below:
Example 2. The following subspace of R2 inherited the Euclidean metric is clearly
locally almost convex:[
X = N × [0, ∞) \{n} × (0, 2−n ).
n∈NHowever, X is not uniformly locally almost convex.
The following are characterizations of uniformly locally almost convexity andalmost convexity:
Theorem 5.1. For a metric space X = (X, d), the following are equivalent:(a) X is uniformly locally almost convex;(b) there exists some δ > 0 such that for each x, y ∈ X with d(x, y) < δ andfor each ε > 0, there is some z ∈ X with d(x, z), d(y, z) < 12 d(x, y) + ε;(c) there exists δ > 0 such that for each 0 < λ < δ and for each x, y ∈ X withd(x, y) < λ, there exist a λ-Lipschitz map f : D → X with f (0) = x andf (1) = y;(d) there is δ > 0 such that N (N (A, s), t) = N (A, s + t) for each A ⊂ X andfor each s, t > 0 with s + t 6 δ.
Theorem 5.2. For a metric space X = (X, d), the following are equivalent:HYPERSPACES WITH THE HAUSDORFF METRIC

63(a) X is almost convex;(b) for each x, y ∈ X and for each ε > 0, there is some z ∈ X with d(x, z),d(y, z) < 12 d(x, y) + ε;(c) for each λ > 0 and for each x, y ∈ X with d(x, y) < λ, there exist a λLipschitz map f : D → X with f (0) = x and f (1) = y;(d) N (N (A, s), t) = N (A, s + t) for each A ⊂ X and for each s, t > 0.
As is easily observed, every uniformly locally almost convex metric space X is
uniformly locally C ∗ -connected. Hence, we have the following:
Corollary 5.3. For every uniformly locally almost convex metric space X, eachcomponent of CldH (X) is a uniform AR and BddH (X) is a uniform ANR.
One should note that the unit circle S1 ⊂ R2 with the Euclidean metric isuniformly locally C ∗ -connected but not uniformly locally almost convex.
Recall a metric space X = (X, d) (or a metric d) is convex if for each x, y ∈ X,
there is some z ∈ X with d(x, z) = d(y, z) = d(x, y)/2. A complete metric space X
is convex if and only if for each x, y ∈ X, there is a map f : [0, d(x, y)] → X withd(x, f (t)) = t. As is easily observed, every almost convex compact metric space isconvex.
Problem 2. Does there exist an almost convex complete metric space which is notconvex?
It is well-know that a Peano continuum2 has an admissible convex metric (cf.
[3]). It is said that X is locally convex if each x ∈ X has a neighborhood U whichis convex. Moreover, X is uniformly locally convex if there is some δ > 0 such thatfor each x, y ∈ X with d(x, y) < δ there is z ∈ X with d(x, z) = d(y, z) = d(x, y)/2.
Problem 3. Does a locally connected metric space possess an admissible metricthat is locally convex, (uniformly) locally almost convex, or uniformly locally Dconnected, uniformly locally C ∗ -connected?
6. The uniformly local C-connectedness
In this section, we show that the uniformly local C ∗ -connectedness is a stronger
condition than the uniformly local version of C-connectedness. It is said that X is
uniformly locally C-connected if for each ε > 0 there exists δ > 0 with the following
property:(ulC) for each η > 0 and each x, y ∈ X with d(x, y) < δ, there is an η-chain(xn )kn=0 in X from x to y with the diameter < ε.
Proposition 6.1. If BddH (X) is uniformly locally path-connected, then X is uniformly locally C-connected.
Every uniformly locally path-connected metric space is uniformly locally Cconnected, but the converse does not hold as the space Q of rationals. Moreover,every uniformly locally C ∗ -connected metric space is uniformly locally C-connected,
but the converse does not hold as the following example:

2A connected compact metrizable space is called a continuum and a locally connected continuum is called a Peano continuum.

64

M. KURIHARA, K. SAKAI, AND M. YAGUCHI

Example 3. For each n ∈ N, let en be the unit vector in RN defined en (i) = 0 if
i 6= n and en (n) = 1. We define a metric space X = (X, d) as follows:[
X
X=
Ren ⊂ RN , d(x, y) =min{2−n , |x(n) − y(n)|}.
n∈N

n∈N

Then, X is uniformly locally C-connected but it is not uniformly locally C ∗ connected.
Remark 1. In Example 3 above, CldH (X) is locally path-connected, hence so is
BddH (X) and they would be ANR’s.
As saw in the above, in order that CldH (X) (or BddH (X)) is an ANR, it is notnecessary that X is uniformly locally C ∗ -connected. The following problems areopen:
Problem 4. When BddH (X) is uniformly locally path-connected (hence it is a
uniform ANR), is X uniformly locally C ∗ -connected?
Problem 5. Does the converse of Proposition 6.1 above hold? Or, for each uniformly locally C-connected metric space X, is CldH (X) (uniformly) locally pathconnected?
It is easy to see that every uniformly locally path-connected metric space isuniformly locally C-connected.
Problem 6. For each uniformly locally path-connected metric space X, is CldH (X)(or BddH (X)) (uniformly) locally path-connected?
Theorem 6.2. Let (X, d) be a uniformly locally C-connected metric space. Then,each x ∈ X has a neighborhood basis consisting of C-connected open neighborhoodof x.
It is known that every compact C-connected set is connected. By using this fact,
we can prove the following lemma.
Lemma 6.3. Every locally compact uniformly locally C-connected metric space X
is locally connected.
Theorem 6.4. Let X = (X, d) be a totally bounded uniformly locally C-connected
metric space. Then, X has a uniformly locally almost convex metric which is uniformly equivalent to d.
By Theorem 6.4 above and Corollary 5.3, we have the following:
Corollary 6.5. For every totally bounded uniformly locally C-connected metricspace X, the space CldH (X) (= BddH (X)) is a uniform ANR.
Problem 7. In Theorem 6.4 and Corollary 6.5, is the total boundedness essential?
Problem 8. For complete metric spaces, does the C-connectedness imply the connectedness?
7. Further problems and related results
The following proposition is shown in [7], which shows the complexity of thespace CldH (X).

HYPERSPACES WITH THE HAUSDORFF METRIC

65

Proposition 7.1. The space CldH (Rn ) has uncountably many components. Moreover, the space of all compact sets CompH (Rn ) is one of them and all but thiscomponent are non-separable.
The following is known.
Theorem 7.2. If a metric space X = (X, d) is complete, then so is every component of CldH (X). Consequently, BddH (X) is complete.
By the main Theorem and Theorem 7.2, for an arbitrary Banach space X, every
component of CldH (X) is a complete metric AR.
Problem 9. For a Banach space (or a Hilbert space) X, is every component of
CldH (X) homeomorphic to a Hilbert space?
Even if X is Euclidean space Rn , the above is unknown, that is,
Problem 10. Is each non-separable component of CldH (Rn ) homeomorphic to a
Hilbert space?
In relation to above problems, some results with different topologies have beenobtained in [1],refer to the bookQ [6], [12] and [13]. For topologies on hyperspaces, we Q[2]. Let Q = i∈N [−2−i , 2−i ] be the Hilbert cube and B(Q) = Q \ i∈N (−2−i , 2−i )be the pseudo-boundary of Q. By `2 (τ ), we denote the Hilbert space with weight τ .
Let `f2 and Qf be the subspaces of the separable Hilbert space `2 and Q respectively,
defined as follows:`f2 = {(xi )i∈N ∈ `2 | xi = 0 except for finitely many i ∈ N}.
Qf = {(xi )i∈N ∈ Q | xi = 0 except for finitely many i ∈ N}.
Theorem 7.3. [13] For a Hausdorff space X, the hyperspace CldF (X) with theFell topology is homeomorphic to Q \ {0} if and only if X is locally compact,
locally connected, separable metrizable and has no compact components, whence
CompF (X) ≈ B(Q). In case X is strongly countable-dimensional, FinF (X) ≈ Qf .
Theorem 7.4. [1], [12] For every infinite-dimensional Banach space X with weightτ , the hyperspace CldAW (X) with Attouch-Wets topology is homeomorphic to `2 (2τ ),FinAW (X) ≈ CompAW (X) ≈ `2 (τ ) × `f2 and
BddAW (X) ≈ `2 (2τ ) × `f2 .
Theorem 7.5. [6] For every infinite-dimensional separable Banach space X, thehyperspace CldW (X) with the Wijsman topology is homeomorphic to `2 and
FinW (X) ≈ BddW (X) ≈ `2 × `f2 .
References[1] T. Banakh, M. Kurihara and K. Sakai, Hyperspaces of normed linear spaces with the AttouchWets topology, Set-Valued Anal. 11(2003), 21-36.
[2] G. Beer, Topologies on Closed and Closed Convex Sets, MIA 268, Kluwer Acad. Publ.,
Dordrecht, 1993.
[3] R. H. Bing Partitioning a set, Bull. Amer. Math. Soc. 55 (1949), 1101–1110.
[4] C. Costantini and W. Kubiś, Paths in hyperspaces, preprint.
[5] Doug Curtis and Nguyen To Nhu, Hyperspaces of finite subsets which are homeomorphic toℵ0 -dimensional linear metric spaces, Topology Appl. 19 (1985), 251–260.

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M. KURIHARA, K. SAKAI, AND M. YAGUCHI[6] W. Kubiś, K. Sakai and M. Yaguchi, Hyperspaces of separable Banach spaces with the Wijsman topology, preprint.
[7] M. Kurihara, Hyperspaces of metric spaces, Master’s Thesis, University of Tsukuba, 2000.
[8] M. Kurihara, K. Sakai and M. Yaguchi, Hyperspaces with the Hausdorff metric and uniform
ANR, preprint.
[9] J.D. Lawson, Topological semilattices with small subsemilattices, J. London Math. Soc. (2)
1 (1969), 719–724.
[10] E. Michael, Uniform AR’s and ANR’s, Compositio Math. 39 (1979), 129–139.
[11] K. Sakai, The comletions of metric ANR’s and homotopy dense subsets, J. Math. Soc. Japan52 (2000), 835–846.
[12] K. Sakai and M. Yaguchi, Hyperspaces of infinite-dimensional Banach spaces with the
Attouch-Wets topology, preprint.
[13] K. Sakai and Z. Yang, Hyperspaces of non-compact metrizable spaces which are homeomorphic to Hilbert cube, Topology Appl. 127(2002), 331-342.
[14] M. WojdysÃlawski Rétractes absoulus et hyperespaces des continus, Fund. Math. 32 (1939),
184–192.
(M. Kurihara) Institute of Mathematics, University of Tsukuba, Tsukuba, 305-8571
JAPANE-mail address: kurihara@first.tsukuba.ac.jp(K. Sakai) Institute of Mathematics, University of Tsukuba, Tsukuba, 305-8571 JAPANE-mail address: sakaiktr@sakura.cc.tsukuba.ac.jp(M. Yaguchi) Institute of Mathematics, University of Tsukuba, Tsukuba, 305-8571
JAPANE-mail address: masato@math.tsukuba.ac.jp

Proceedings of General Topology Symposium, p. 67 - p. 72, 2002

http://gt2002.h.kobe-u.ac.jp/

APPROXIMATE RESOLUTIONS AND AN APPLICATION TO
HAUSDORFF DIMENSIONTAKAHISA MIYATA AND TADASHI WATANABE
Abstract. In this paper, we present a new approach using normal sequences
and approximate inverse systems to study Hausdorff dimension for compact
metrizable spaces.

This short article summarizes the main results from the paper [8].
1. Preliminary
For each subset F of Rm and for each s ≥ 0, the s-dimensional Hausdorff measureof F is defined as Hs (F ) = lim Hsδ (F ) where for each δ > 0,
δ→0Hsδ (F ) = inf∞
X|Ui |si=1where the infimum is taken over all open balls Ui with radius at most δ such that∞F ⊆ ∪ Ui . Here |Ui | denotes the diameter of the set Ui . The Hausdorff dimension
i=1

of F is defined as dimH F = sup{s : Hs (F ) = ∞} (= inf{s : Hs (F ) = 0}) [2]. The
present paper concerns Hausdorff dimension for non-Euclidean spaces. More precisely, we develop a systematic approach using normal sequences and approximate
inverse systems to study Hausdorff dimension for compact metrizable spaces.
Throughout the paper, all spaces are assumed to be metrizable, and maps meancontinuous maps.
For any space X, let Cov(X) denote the family of all open coverings of X. For
any U, V ∈ Cov(X), U is a refinement of V, in notation, U < V, if for each U ∈ U
there is V ∈ V such that U ⊆ V . For any subset A of X and U ∈ Cov(X), letst(A, U) = ∪{U ∈ U : U ∩ A 6= ∅} and U|A = {U ∩ A : U ∈ U}. If A = {x}, we writest(x, U) for st({x}, U). For each U ∈ Cov(X), let st U = {st(U, U) : U ∈ U}. Let
st1 U = st U and stn+1 U = st(stn U) for each n ∈ N. For any metric space (X, d)and r > 0, let Ud (x, r) = {y ∈ X : d(x, y) < r}, and for each subset A of X, let |A|denote the diameter of A. For any U ∈ Cov(X), two points x, x0 ∈ X are U-near,denoted (x, x0 ) < U, provided x, x0 ∈ U for some U ∈ U. For any V ∈ Cov(Y ),two maps f, g : X → Y between spaces are V-near, denoted (f, g) < V, provided(f (x), g(x)) < V for each x ∈ X. For each U ∈ Cov(X) and V ∈ Cov(Y ), let1991 Mathematics Subject Classification. Primay. 54E35, 54E40, Secondary. 28A78, 28A80,
54F45, 54C56 .
Key words and phrases. Approximate resolution, normal sequence, Hausdorff dimension, Cantor set.
67

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T. MIYATA, AND T. WATANABE

f (U) = {f (U ) : U ∈ U} and f −1 (V) = {f −1 (V ) : V ∈ V}. Let I denote the closed
interval [0, 1], and let N denote the set of all positive integers.
Metrics induced by normal sequences. A family U = {Ui : i ∈ N} of open
coverings on a space X is said to be a normal sequence provided st Ui+1 < Ui foreach i. Let ΣU denote the normal sequence {Vi : Vi = Ui+1 , i ∈ N} and st U thenormal sequence {st Ui : i ∈ N}. For any normal sequences U = {Ui } and V = {Vi },we write U < V provided Ui < Vi for each i. For each map f : X → Y and foreach normal sequence V = {Vi }, let f −1 (V) = {f −1 (Vi )}. For each subspace A of
X, let U|A denote the normal sequence {Ui |A} on A.
Following the approach by Alexandroff and Urysohn, given a space X and anormal sequence U = {Ui } on X, we define the metric dU on X as follows (for moredetails, see [6]):dU (x, x0 ) = inf{DU (x, x1 ) + DU (x1 , x2 ) + · · · + DU (xn , x0 )}where the infimum is taken over all points x1 , x2 , ..., xn in X, andif (x, x0 ) 6< U1 ;
 9
0
1
if (x, x0 ) < Ui but (x, x0 ) 6< Ui+1 ;DU (x, x ) =
i−2
 3
0
if (x, x0 ) < Ui for all i ∈ N ,Then the metric dU has the property1) ⊆ st(x, Ui ) for each x ∈ X and i.
3i
In particular, if U = {Ui } is the normal sequence on a metric space (X, d) suchthat Ui = {Ud (x, 31i ) : x ∈ X}, then the metric dU induces the uniformity which isisomorphic to that induced by the metric d. Moreover, if X is a convex subset of alinear toplogical space, then dU is isometric to the original metric.
st(x, Ui+3 ) ⊆ UdU (x,Proposition 1. Let U = {Ui } and V = {Vi } be normal sequences on X. Then for
all x, x0 ∈ X,
(1) if U < V, then dU (x, x0 ) ≥ dV (x, x0 ),(2) dΣU (x, x0 ) = 3 dU (x, x0 ),(3) dst U (x, x0 ) ≤ dU (x, x0 ) ≤ 3 dst U (x, x0 ).
Approximate sequences and resolutions. An inverse sequence (Xi , pi,i+1 )consists of spaces Xi , called coordinate spaces, and maps pi,i+1 : Xi+1 → Xi . We
write pij for the composite pi,i+1 pi+1,i+2 · · · pj−1,j if i < j, and let pii = 1Xi ,and call maps pij bonding maps. An approximate inverse sequence (approximate
sequence, in short) X = (Xi , Ui , pi,i+1 ) consists of an inverse sequence (Xi , pi,i+1 )and Ui ∈ Cov(Xi ) and must satisfy the following condition:(AI): For each i ∈ N and U ∈ Cov(Xi ), there exists i0 > i such that Ui00 <00
0p−1
ii00 U for i > i .
An approximate map p = (pi ) : X → X of a compact space X into an approximate
sequence X = (Xi , Ui , pi,i+1 ) consists of maps pi : X → Xi for i ∈ N, called
projection maps, such that pi = pij pj for i < j, and it is an approximate resolutionof X if it satisfies the following two conditions:(R1): For each ANR P , V ∈ Cov(P ) and map f : X → P , there exist i ∈ N
and a map g : Xi → P such that (gpi , f ) < V, andAPPROXIMATE RESOLUTIONS AND HAUSDORFF DIMENSION

69(R2): For each ANR P and V ∈ Cov(P ), there exists V0 ∈ Cov(P ) suchthat whenever i ∈ N and g, g 0 : Xi → P are maps with (gpi , g 0 pi ) < V0 ,then (gpii0 , g 0 pii0 ) < V for some i0 > i.
The following is a useful characterization:
Theorem 2. ([5, Theorem 2.8]) An approximate map p = (pi ) : X → X =(Xi , Ui , pi,i+1 ) is an approximate resolution of X if and only if it satisfies the following two conditions:(B1): For each U ∈ Cov(X), there exists i0 ∈ N such that p−1
i Ui < U fori > i0 , and(B2): For each i ∈ N and U ∈ Cov(Xi ), there exists i0 > i such thatpii0 (Xi0 ) ⊆ st(pi (X), U) for i0 > i0 .
An approximate resolution p = (pi ) : X → X is said to be normal if the family
U = {p−1
i (Ui )} is a normal sequence. Then each normal approximate resolution p
induces a metric dU , which will be denoted by dp .
Theorem 3 ([5]). Every compact space X admits a normal approximate resolutionp = (pi ) : X → X = (Xi , Ui , pi,i+1 ) such that all coordinate spaces Xi are finite
polyhedra.
Throughout the paper, every normal approximate resolution is assumed to havethe property of Theorem 3. Our notions of approximate systems and approximate
resolutions are the commutative versions of the corresponding notions in [4] and[5].
2. Hausdorff dimension
Let X be a σ-compact space with a normal sequence U = {Ui }. For each s ≥ 0,for each subset F of X and for each i ∈ N, we define(∞ µ)∞
X 1 ¶s[s
HU,i (F ) = inf:F ⊆Uik , Uik ∈ Uik , i ≤ ik ,
3ik
k=1

and

k=1

HsU (F ) = lim HsU,i (F ).

i→∞¡ 1 ¢t−s st
HU,i (F ) for s < t and for all i, and hence if HsU (F ) < ∞,Then HU,i (F ) ≤ 3itthen HU (F ) = 0 for t > s. Thus we can define the Hausdorff dimension dimU
HFss
of F with respect to U by dimU
H F = inf{s : HU (F ) = 0} (= sup{s : HU (F ) = ∞}).

We can prove

Theorem 4. HsU is a metric outer measure on X with respect to the metric dU .
Hence HsU defines a measure on the Borel subsets of X, which we call the sdimensional Hausdorff measure with respect to U (or s-dimensional Hausdorff Umeasure) on X.
Our Hausdorff dimension coincides with the usual Hausdorff dimension for Euclidean space with a particular normal sequence.
Theorem 5. Let B = {Bi } be the normal sequence on Rn which consists of theopen coverings Bi by open balls with radius 31i . Then for any subset F of Rn andfor each i and s, HsB,i (F ) = 2s Hs1 (F ), and hence dimBH F = dimH F .
3i

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T. MIYATA, AND T. WATANABE

We have the following properties:
Theorem 6. Let U = {Ui } and V = {Vi } be normal sequences on X, and let F be
any subset of X.
V(1) If V < U, then HsU (F ) ≤ HsV (F ), and hence dimU
H F ≤ dimH F .
ΣU
U
s
s s
(2) HΣU (F ) = 3 HU (F ), and hence dimH F = dimH F .
UU(3) Hsst U (F ) ≤ HsU (F ) ≤ 3s Hsst U (F ), and hence dimst
H F = dimH F .
ss(4) (Subset theorem) If F1 ⊆ F2 ⊆ X, then HU (F1 ) ≤ HU (F2 ) and henceU
dimU
H F1 ≤ dimH F2 .
UU(5) (Sum theorem) dimU
H (F1 ∪ F2 ) = max{dimH F1 , dimH F2 } for any subsets
F1 , F2 of X.

For any spaces X and Y with normal sequences U = {Ui } and V = {Vi },respectively, a map f : X → Y is called a (U, V)-Lipschitz map provided there
exists a constant α > 0 such thatdV (f (x), f (x0 )) ≤ α dU (x, x0 ) for x, x0 ∈ X,
and a (U, V)-bi-Lipschitz map provided there exist constants α1 , α2 > 0 such that
α1 dU (x, x0 ) ≤ dV (f (x), f (x0 )) ≤ α2 dU (x, x0 ) for x, x0 ∈ X.
Lipschitz maps and bi-Lipschitz maps are characterized in terms of normal sequences as follows:
Theorem 7. ([6, §.5, §.7] and [7, §.3]) Consider the following conditions:(L)k : dV (f (x), f (x0 )) ≤ 3k dU (x, x0 ) for x, x0 ∈ X,(L)k : dU (x, x0 ) ≤ 3k dV (f (x), f (x0 )) for x, x0 ∈ X,(N)m,n : Σm U < f −1 (Σn V),(N)m,n : f −1 (Σm V) < Σn U.
Then for m, n ≥ 0,(1) (N)m,n ⇒ (L)n−m ; (L)m ⇒ (N)m+4,0 ; (L)−m ⇒ (N)4,m , and(2) if f is surjective, then (L)m ⇒ (N)m+4,0 ; (L)−m ⇒ (N)4,m ; (N)m,n ⇒(L)m−n .
We have the Lipschitz invariance:
Theorem 8. Let f : X → Y be a map between σ-compact spaces X and Y withnormal sequences U = {Ui } and V = {Vi }, respectively, and let F be a subset of X.
For m ≥ 0, consider the properties:(H)m : HsV (f (F )) ≤ 3ms HsU (F ) for s ≥ 0, and(H)m : HsU (F ) ≤ 3ms HsV (f (F )) for s ≥ 0.
Then for m ≥ 0,
U(1) (L)m ⇒ (H)m+4 ⇒ dimV
H f (F ) ≤ dimH F .
V
U
m
m+4
(2) (L) ⇒ (H)⇒ dimH f (F ) ≥ dimH F .

Corollary 9. If f : X → Y is a (U, V)-bi-Lipschitz map. Then for any subset F
Uof X, Then dimV
H f (F ) = dimH F .

APPROXIMATE RESOLUTIONS AND HAUSDORFF DIMENSION

71

3. An inverse system approach
If p = (pi ) : X → X = (Xi , Ui , pi,i+1 ) is an approximate resolution of a compact
space X, for each s ≥ 0 and i ∈ N, we define Hsi (p) as
( n µ
)
n
X 1 ¶s[inf: pj (X) ⊆pik j (Uik ), Uik ∈ Uik , i ≤ ik ≤ j, n ∈ N
3i k
k=1

k=1

and define the s-dimensional Hausdorff measure of p as Hs (p) = lim Hsi (p). Simi→∞

ilarly to dimU
H , we can define the Hausdorff dimension of p as dimH (p) = sup{s :Hs (p) = ∞}.
Lemma 10. Let p = (pi ) : X → X = (Xi , Ui , pi,i+1 ) be a normal approximatessresolution, and let U = {p−1i (Ui )}. Then for each s ≥ 0, H (p) = HU (X).
For each approximate sequence X = (Xi , Ui , pi,i+1 ) and for each i, we define
Hsi (X) as
)
( n µ
n
[
X 1 ¶s−1: Xj ⊆
pik j (Uik ), Uik ∈ Uik , i ≤ ik ≤ j, n ∈ N
inf
3ik
k=1

k=1

and define the s-dimensional Hausdorff measure of X as Hs (X) = lim Hsi (X).
i→∞

Similarly to dimH (p), we can define the Hausdorff dimension dimH X of X as
dimH X = sup{s : Hs (X) = ∞}. Note here that the definition of Hsi (X) does notdepend on the projection maps pi .
Lemma 11. Let p = (pi ) : X → X = (Xi , Ui , pi,i+1 ) be a normal approximate
resolution such that(1)

st Uj < p−1
ij Ui for i < j,and let F be a compact subset of X. For each i, let Fi be a compact polyhedronsuch thatst(pi (F ), Ui ) ⊆ Fi ⊆ st(pi (F ), st Ui ).
Then(1) F = (Fi , Ui |Fi , pi,i+1 |Fi+1 ) is an approximate sequence, and p|F = (pi |F ) :F → F = (Fi , Ui |Fi , pi,i+1 |Fi+1 ) is a normal approximate resolution; andss(2) if U = {p−1i (Ui )}, then HU (F ) = H (F ) for each s ≥ 0.
Remark. Given any normal approximate resolution p : X → X of X, we can always
find a normal approximate resolution p0 : X → X 0 of X so that X 0 is a subsystem
of X and has property (1).
By Lemmas 11 and 10, we have characterizations of dimU
H in terms of an approximate sequence and in terms of an approximate resolution.
Theorem 12. Under the same setting as in Lemma 11, dimH F = dimU
HF =dimH (p|F ).
Remark. All the results in this section hold for the noncommutative versions ofapproximate sequences and approximate resolutions (limits) in the sense of [4, 5].
It is well-known that if X is a compact metrizable space with covering dimension
n, then X can be embedded in I2n+1 [3, Theorem V 2]. Motivated by this result,

72

T. MIYATA, AND T. WATANABE

we consider the following question: For each r > 0, find the least integer N for
which a Cantor set with Hausdorff dimension r can be realized in the cube [0, 1]N .
N
For each N ∈ N and for each i ∈ N, let IN
with the usual metric d, let Ui
i = I
N
1
be the open covering by open 3i+1
-balls, and let qi,i+1 : IN
i+1 → Ii be the identity
NN
map. Then it is easy to see that I = (Ii , Ui , qi,i+1 ) is an approximate sequence.
For each i ≥ 1, let qi : IN → INi be the identity map. Then the approximate map
q = (qi ) : IN → I N is a normal approximate resolution of IN , and the metric dq
induced by q is isometric to the metric d on IN .
Theorem 13. For each positive real number r, let
·
¸
log 3
N=
(r + 1) + 1 .
log 2Then there exist a Cantor set X in IN and compact subsets Xi of IN
i so that the
restriction p = (qi |X) : X → X = (Xi , Ui |Xi , qi,i+1 |Xi+1 ) is a normal approximate
resolution of X, and dimH (p) = r. Here, for each r > 0, let [r] denote the least
integer that is greater than or equal to r.
References[1] P. Alexandroff and P. Urysohn, Une condition nécessaireet suffisante our qu’une classe (L)soit une classe (B), C. R. Acad. Sci. Paris Sér. A-B 177(1923), 1274 - 1276.
[2] K. Falconer, The geometry of fractal sets, Cambridge Univ. Press, Cambridge, 1985.
[3] W. Hurewicz and H. Wallman, Dimension theory, Princeton Univ. Press, 1948.
[4] S. Mardešić and L. R. Rubin, Approximate inverse systems of compacta and covering dimension, Pacific J. Math. 138 (1989), 129 - 144.
[5] S. Mardešić and T. Watanabe, Approximate resolutions of spaces and maps, Glasnik Mat.
24 (1989), 587 - 637.
[6] T. Miyata and T. Watanabe, Approximate resolutions and Lipschitz functions, Top. Appl.
122(2002), 353 - 375.
[7] T. Miyata and T. Watanabe, Bi-Lipschitz maps and the category of approximate resolutions,
Glasnik Mat. (to appear).
[8] T. Miyata and T. Watanabe, Approximate inverse systems and Hausdorff dimension,
preprint.
[9] K. Nagami, Dimension theory, Academic Press, New York, 1970.
[10] T. Watanabe, Approximative shape I, Tsukuba J. Math. 11 (1987), 17 - 59.
(T. Miyata) Division of Mathematics and Informatics, Faculty of Human Development,
Kobe University, 3-11 Tsurukabuto, Nada-Ku, Kobe, 657-8501 JAPANE-mail address, T. Miyata: tmiyata@kobe-u.ac.jp(T. Watanabe) Department of Mathematics and Information Sciences, Faculty of Education, Yamaguchi University, Yamaguchi-City, 753-8513 JAPANE-mail address, T. Watanabe: tadashi@po.yb.cc.yamaguchi-u.ac.jp

Proceedings of General Topology Symposium, p. 73 - p. 77, 2002

http://gt2002.h.kobe-u.ac.jp/

ON STRONGLY σ-SHORT BOOLEAN ALGEBRASMAKOTO TAKAHASHI
Abstract. We investigate strongly σ-shortness of some Boolean algebras. Especially, we show that every (κ, ω)-caliber Boolean algebra of density ≥ κ isnot strongly σ-short.

1. Introduction
In [5], we introduced σ-shortness and strongly σ-shortness of Boolean algebras.
We say that a subset D of a Boolean algebra B is σ-short if every strictly descending
sequence of length ω in D does not have a nonzero lower bound in B, ∧-closed if
d1 ∧ d2 ∈ D for every d1 , d2 ∈ D such that d1 ∧ d2 > 0. B is said to be σ-short
if it has a σ-short dense subset and strongly σ-short if it has a σ-short ∧-closed
dense subset. We note that B itself is not a σ-short set if B is atomless. In thispaper, we assume that Boolean algebras are atomless. Typical examples of σ-short
Boolean algebras are regularly filtered Boolean algebras which are also strongly
σ-short. Another examples of σ-short Boolean algebras are measure algebras. In[5], we left the following problems:(1) Are measure algebras strongly σ-short?
(2) Is Prikry forcing σ-short?
After the author lectured in the 2002 General Topology Symposium, Jörg Brendle showed the following theorem which concerns to the first problem.
Theorem A(Brendle). Let Bκ be the algebra for adding κ many random reals.
(1) Bω is not strongly σ-short.
(2) Assume that the density of Bκ equals to κ. Then Bκ is strongly σ–short.
Yasuo Yoshinobu and the author extend the first result more general as follows.
Theorem 1. Suppose that B satisfies (κ, ω)-caliber and d(B) ≥ κ where d(B)denotes the density of B. Then B is not strongly σ-short.
Let κ be a measurable cardinal, and U a normal measure on κ. Let PU denotethe canonical poset of the Prikry forcing associated with U and BU be the Boolean
completion of PU . Since BU satisfies (κ, ω)–caliber and d(BU ) ≥ κ, BU is notstrongly σ-short by virtue of the thorem above. Y. Yoshinobu also show that
Prikry forcing itself is not σ-short. However, it is still open whether Prikry algebra
BU is not σ-short.
I would like to thank Jörg Brendle and Yasuo Yoshinobu for many valuablecomments. In this paper we give proofs of results above with their permission.
1991 Mathematics Subject Classification. 03G05, 06E05.
Key words and phrases. Boolean algebras; σ-short; strongly σ-short.
73

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MAKOTO TAKAHASHI

2. Preliminaries
In this section we give basic definitions, notation and results which are needed inthis paper. The reader is assumed to be familiar with the basic facts about Boolean
algebras found in [3]. For basic facts about set theory, we refer to [2] and [4].
We use the letter κ for infinite cardinals; the letters α, β for ordinals; the letters
A, B for infinite atomless Boolean algebras. For a Boolean algebra B, we denoteby B+ the set of all nonzero elements of B. We use ∧, ∨, − for Boolean operations.
A ≤ B means that A is a subalgebra of B. If X ⊂ A, then hXiA is the subalgebraof A generated by X. We omit the subscript if there is no confusion. We say thata set D ⊆ B+ is dense if for every b ∈ B+ there exists d ∈ D such that d ≤ b.
For a poset P, we denote by B(P) the Boolean completion of P. For a set X, wedenote by |X| the cardinality of X.
¯ For a Boolean algebra B, we define the densityd(A) of B by d(A) =¯ min{|X| ¯ XisdenseinB}. For a set X and a cardinal κ,let [X]κ = {Y ⊆ X ¯ |Y | = κ}. C ⊆ [X]κ is said to be λ-closed if it is closedwith respect to union of increasing sequences of length ≤ κ. C ⊆ [X]κ is said tobe unbounded if it is cofinal in ([X]κ , ⊆). We say that a Boolean algebra B has(κ, ω)-caliber if for any uncountable subset T ⊆ B of size κ, there is countableF ⊆ T such that F has a non-zero lower bound in B. It is well-known that the
random algebra has (ω1 , ω)-caliber.
A is called a regular subalgebra of B, in symbol A ≤reg B, whenever for every
VA
VB
M ⊆ A, ¯ M = 0 implies
M = 0. B is said to be regularly filtered if{A ≤reg B ¯ |A| ≤ ℵ0 } contains an ℵ0 -closed unbounded subset of [B]ℵ0 . Basicfacts about regularly filtered Boolean algebras can be found in [1]3. Examples
In this section, we shall give some examples of σ-short Boolean algebras.
Example 1. For any set X, let Fr X be the free Boolean algebra over X. We
assume without loss of generality that X ⊂ Fr X. Put¯
D = {±x1 · ±x2 · . . . · ±xn ¯ n ∈ ω, x1 , x2 , · · · , xn ∈ X} − {0}.
Clearly, D is a σ-short ∧-closed dense subset of Fr X. Hence, Fr X is strongly
σ-short.
Example 2. Let (B, µ) be a measure algebra. Put¯
1for some n ∈ ω}.
D = {a ∈ B ¯ µ(a) =n+1
Then D is a σ-short dense subset of B. Hence (B, µ) is σ-short. (B, µ) is not
regularly filtered. In fact, measure algebras are weakly σ-distributive but free
Boolean algebras are not weakly σ-distributive, and a regular subalgebra of a weakly
σ-distributive Boolean algebra is again weakly σ-distributive. Thus (B, µ) does nothave any countable atomless regular subalgebra.
Example 3. Let (P, ≤) be a notion of forcing with finite conditions such that p ≥ q
if and only if p ⊂ q for every p, q ∈ P. P itself is σ-short and is a σ-short dense
subset of B(P). Moreover, if any two compatible elements of P have an infimumin P, then P is ∧-closed in B(P), so that B(P) is strongly σ-short.
Every regularly filtered Boolean algebra is strongly σ-short(see [5]). The following example, however, shows that the converse is not true even if it is ccc.

ON STRONGLY σ-SHORT BOOLEAN ALGEBRAS

75

Example 4. Let S ⊂ ω1 be a stationary co-stationary set and (T, <T ) be the treeof closed subset of S under end extension. Let PT be the set of all finite antichainsof T ordered by reversed inclusion. Then PT is a σ-short ∧-closed dense subsetof B(PT ) by Example 3, so that B(PT ) is strongly σ-short. PT is ccc but not
absolutely ccc(see [6]). Since every regularly filtered Boolean algebra is absolutely
ccc, B(PT ) is not regularly filtered.
4. strongly σ-short Boolean algebras
In this section, we prove main results.
Theorem 1. Suppose that B satisfies (κ, ω)-caliber and d(B) ≥ κ. Then B is notstrongly σ-short.
Proof. Suppose that B is strongly σ-short. Let D be a dense, σ-short and ∧–closed
subset of B. There is D0 ⊆ D dense, |D0 | = d(B), which is still σ-short and∧-closed. Enumerate D0 = {dα ; α < d(B)}. We recursively construct sets D1α(α < d(B))and X such that
(1) D1α ⊆ D1β for α < β(2) |D1α | ≤ |α| · ω < d(B)(3) D1α ⊆ D0 is ∧–closed(4) for α < β, if d ∈ D1α and e ∈ D1β \ D1α , then e 6≥ d.
(5) ∀α < d(B)∃x ∈ D1α+1 [x ≤ dα ](6) ∀α < d(B), α ∈ X if and only if there is no d ∈ D1α such that dα ≥ dS
Let D10 = ∅ and X0 = ∅. For a limit ordinal λ, let D1λ = α<λ D1α and Xλ =Sβα<λ Xα . Let α = β + 1 be a successor ordinal. If there is d ∈ D1 such thatββdβ ≥ d, let D1α = D1 and Xα = Xβ . If there is no d ∈ D1 such that dβ ≥ d, letS
D1α be the ∧–closure of D1β ∪ {dβ } and Xα = Xβ ∪ {β}. Put X = α<d(B) Xα andS
D1 = α<d(B) D1α . Since D1 ⊆ D0 , D1 is σ-short. It is easy to see that D1 is denseand ∧–closed.
Since d(B) ≥ κ and X is cofinal in d(B), |X| ≥ κ. Since B satisfies (κ, ω)caliber, there exists countable subset F of X such that {dα |α ∈ F } has a non-zero
lower bound b in B. Without loss of generality, we assume F = {αn |n ∈ ω} andα1 < α2 < · · · < αn < · · · . Since αn ∈ X, dαn 6≥ dα1 ∧ dα2 ∧ ... ∧ dαn−1 . Hence thesequence of en = dα1 ∧ dα2 ∧ ... ∧ dαn ∈ D1 is strictly decreasing. Since dαn ≥ b for
every n ∈ ω, {en |n ∈ ω} has a non-zero lower bound B. So D1 is not σ-short. This
contradicts that D0 is σ-short.
¤
Let Bκ be the algebra for adding κ many random reals. Since Bω satisfies(ω1 , ω)-caliber and d(Bω ) ≥ ω1 , we have the following:
Corollary 1 (Brendle). Bω is not strongly σ-short.
On the other hand, J. Brendle also showed that
Theorem 2 (Brendle). Assume that d(Bκ ) = κ. Then Bκ is strongly σ–short.
Proof. Let D ⊆ Bκ be dense, |D| = κ. Say D = {bα ; α < κ}. For each α choose
γα ∈/ supp(bα ) in such a way that the γα are distinct for distinct α. Let aα =bα ∧ [{hhγα , 0i, 0i}]. Here {hhγα , 0i, 0i} denotes the partial function p : κ × ω → 2
with domain the singleton {hγα , 0i} and p(hγα , 0i) = 0. [p] is the open set definedby p. Let D0 be the ∧–closure of the collection of the aα . Assume {dn ; n ∈ ω} ⊆ D0

76

MAKOTO TAKAHASHI

V
is strictly decreasing. Each dn is a finite conjunction of aα , say dn = i<kn aαi,n . IfV{aαi,n ; i < kn , n ∈ ω} is infinite, n dn = 0 is immediate. If {aαi,n ; i < kn , n ∈ ω}
is finite, then the sequence cannot be strictly decreasing, a contradiction. So D0 is
σ–short.
¤Let κ be a measurable cardinal, and U a normal measure on κ. Let PU denotethe canonical poset of the Prikry forcing associated with U , that is, PU consists ofall pairs (s, A) satisfying(1) s is a finite strictly increasing sequence of ordinals below κ,(2) A ∈ U and max s < min A (consider max ∅ = −1),and (s, A) ≤ (t, B) iff(1) s is an end extension of t,(2) s \ t ⊆ B, and A ⊆ B.
Let B be the Boolean completion of PU . Since B satisfies (κ, ω)–caliber andd(B) ≥ κ, B is not strongly σ-short. It is open whether B is σ-short. However, Y.
Yoshinobu showed that PU itself is not σ-short.
Claim 1. Whenever D is a dense subset of PU , there exists s such that∀X ∈ U ∃Y ∈ U [Y ⊆ X ∧ (s, Y ) ∈ D].
Proof. Suppose not. Then for every s there is Xs ∈ U such that∀Y ∈ U [Y ⊆ Xs → (s, Y ) ∈/ D].
For each −1 ≤ α < κ, set
Xα :=

\

Xs

max s=αand
X := 4−1≤α<κ Xα .
Note that X ∈ U and thus (∅, X) ∈ PU . Therefore there is (s, Y ) ∈ D such that(s, Y ) ≤ (∅, X). But then for every α ∈ Y , α ∈ Xmax s ⊆ Xs and thus Y ⊆ Xs
holds. Contradiction.
¤
Theorem 3. Prikry forcing is not σ-short.
Proof. Pick s as in the above claim, and pick X0 ∈ U such that (s, X0 ) ∈ D.
Whenever Xn is given, pick Xn+1 ∈ U such that Xn+1 ⊆ Xn \ {min Xn } and(s, Xn+1 ) ∈ D. Then (s, Xn )’s form a strictly decreasing ω-sequence in D which
has a common extension. This shows that D is not σ-short.
¤
References
1. L. Heindorf and L. B. Shapiro, Nearly Projective Boolean Algebras, Lecture Notes in Math.
1596, Springer-Verlag, 1994.
2. T. Jech, Set Theory, Academic Press, 1978.
3. S. Koppelberg, General Theory of Boolean Algebras, in: Handbook of Boolean Algebras,
J. D. Monk with R. Bonnet(eds.) vol. 1, North-Holland, 1989
4. K. Kunen, Set Theory, North-Holland, 1980
5. M. Takahashi and Y. Yoshinobu, σ-short Boolean algebras, to appear in Mathematical Logic
Quarterly
6. J. Steprans and S. Watson, Destroying precaliber ℵ1 : an application of a ∆-system lemma
for closed sets, Fund. Math. 129 (1988), 223–229.

ON STRONGLY σ-SHORT BOOLEAN ALGEBRAS

77

E-mail address: makoto@kobe-u.ac.jp
Faculty of Human Development, Kobe University, Tsurukabuto, Nada, Kobe 6578501, Japan

Proceedings of General Topology Symposium, p. 79 - p. 89, 2002

http://gt2002.h.kobe-u.ac.jp/

SMOOTHNESS OF SCALING FUNCTIONS AND TRANSFEROPERATORSTATSUHIKO YAGASAKI

この論説は，rank M 直交 スケール関数 と それに付随する ウェーブレット関数の滑らかさ を Transfer operator の spectral radius を用いて評価するという 米谷・
中岡・大倉氏 との共同研究 [6] の紹介を目的としたものである．

1. ウェーブレット解析について
1.1. Fourier 解析 から Wavelet 解析 へ.
Fourier 解析 から Wavelet 解析 への移行は，数学的には 次の様に理解される[5]. (複素) 関数空間 L2 ([0, 2π]) は，直交基底 einx (n ∈ Z) を持ち，任意の関数f ∈ L2 ([0, 2π]) は Fourier 級数展開
X
f (x) =
cn einxn∈Zを持つ．Fourier 係数 cn は

1
cn =
2π
で与えられる．また，Fourier 変換
れる：

Z 2π

f (x)e−inx dx.

0

F : L2 ([0, 2π]) → L2 ([0, 2π])Z ∞F(f )(ξ) =が次式で定義さf (x)e−iξx dx.

−∞

基底 {einx } の特徴は，１つの基本関数 eix の dilation によって構成されていることである．これに対し，基本的な関数空間である L2 (R) も このような基本関数 ψ(x)を持つかどうか が自然な問題として生じる．R は (非コンパクト) 線形空間 であるから，基本関数に対する変換として，dilation に加えて translation x + k も考慮に入れるのは自然なことである．dilation を 整数倍 nx ではなく 2 の整数乗倍 2j x にとると，問題は 次の様に述べられる：問題 1.1. 関数 ψ(x) ∈ L2 (R) で，ψj,k (x) = 2j/2 ψ(2j x − k) (j, k ∈ Z) が L2 (R) の正規直交基底となるものが存在するか？
この条件を満たす基本関数 ψ(x) を 2 進 直交 ウェーブレット と呼ぶ．ψ(x) はコ
ンパクト台を持つことが望ましい．最も簡単な例は，次で定義される Harr 関数 である：
 1 (0 ≤ x < 1/2)
ψH (x) = −1 (1/2 ≤ x < 1)
0 (その他).
79

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TATSUHIKO YAGASAKI

ψ(x) が 2 進 直交 ウェーブレット のとき，関数 f ∈ L2 (R) に対し，展開X
cj,k ψj,kf=
j,k∈Z

を f の ウェーブレット展開と呼ぶ．ウェーブレット係数 cj,k は，次式で与えられる：
Ã
!
Z ∞
Z ∞
k
x
−
1
j
2
cj,k =
f (x)ψj,k (x) dx = ( 21j )− 2
f (x)ψ
dx.
1
−∞

µ

−∞

¶

2j

k 1
を 連続変数 (b, a) で置き換えて ウェーブレット変換 Wψ : L2 (R) → L2 (R),2j 2jが次式で定義される：
µ
¶
Z ∞
x−b
− 12
Wψ (f )(b, a) = |a|f (x)ψ
dx.
a−∞1.2. 多重解像度解析.
各 j ∈ Z に対して L2 (R) の閉部分空間 Wj を
Wj := clL2 (R) hψj,k : k ∈ Zi
で定めると，次が成り立つ：

L2 (R) = ⊕j∈Z Wj(直交直和).

さらに Vj := ⊕i≤j−1 Wi と置くと，{Vj } は次の条件を満たす：(1) Vj ⊂ Vj+1 (j ∈ Z)(2) clL2 (R) (∪j∈Z Vj ) = L2 (R)(3) ∩j∈Z Vj = {0}(4) f (x) ∈ Vj ⇐⇒ f (2x) ∈ Vj+1 (j ∈ Z)
この条件を満たす閉部分空間の列 {Vj } を 一般に 2 進 直交 MRA (多重解像度解析)と呼ぶ．
逆に，このような 直交 MRA {Vj } を基に，ウェーブレット ψ(x) を構成するという観点に立つと，次の条件を満たす関数 ϕ(x) ∈ L2 (R) の存在を要請することは自然である：(∗) ϕ(x − k) (k ∈ Z) は V0 の正規直交基底．
この条件を満たす関数 ϕ(x) を MRA {Vj } を生成する 2 進 直交 スケール関数 と呼ぶ．ϕj,k (x) = 2j/2 ϕ(2j x − k) (j, k ∈ Z) と置くと，(4) より ϕj,k (x) (k ∈ Z) は Vjの正規直交基底 になる．また，ϕ1,k (k ∈ Z) は V1 の直交基底で，ϕ ∈ V0 ⊂ V1 であるから，ある {ak } ∈ `2 が存在してX
ϕ(x) =ak ϕ(2x − k)k∈Zと一意に表される．この関係式を 2 - スケール関係 と呼ぶ．
2 進 直交 ウェーブレットの具体的な構成では，スケール関数 と ウェーブレット
が組として 構成される．従って，上の整合性を持つ 2 進 直交 MRA {Vj }, スケール関数 ϕ(x), ウェーブレット ψ(x) は，１つの系として捉えられるものである．さらに，2 進条件は，一般の M ≥ 2 に対して M 進条件 に自然に拡張される．
以上の考察に基づいて，一般の M ≥ 2 に対して M 進 直交 MRA の枠組みの中で，M 進 直交 スケール関数 ϕ(x) 及び M 進 直交 ウェーブレット系 ψ 1 (x), · · · , ψ M −1 (x)が，次の様に定義される：

SMOOTHNESS OF SCALING FUNCTIONS AND TRANSFER OPERATORS81

定義 1.1. 次の条件を満たす L2 (R) の閉部分空間の列 {Vj } を M 進 直交 MRA と呼ぶ：(1) Vj ⊂ Vj+1 (j ∈ Z)(2) clL2 (R) (∪j∈Z Vj ) = L2 (R)(3) ∩j∈Z Vj = {0}(4) f (x) ∈ Vj ⇐⇒ f (M x) ∈ Vj+1 (j ∈ Z)(5) ある関数 ϕ(x) ∈ V0 が存在して ϕ(x − k) (k ∈ Z) は V0 の正規直交基底．
−1iii(6) 直交直和分解 V1 = V0 ⊕(⊕M
i=1 W1 ) 及び 関数 ψ (x) ∈ W1 (i = 1, · · · , M −1)iが存在して，各 i = 1, · · · , M − 1 に対して ψ (x − k) (k ∈ Z) は W1i の正規直交基底．
関数 ϕ(x) 及び ψ 1 (x), · · · , ψ M −1 (x) を各々，この MRA における (M 進 直交) ス
ケール関数 及び ウェーブレット系 と呼ぶ．
この定義における 直交基底 という条件は Riesz 基底 という条件に一般化することができる．また，ウェーブレット変換 Wψ では，必要に応じて さらに弱い条件の下で 基本関数 ψ を選ぶことができる．
このように定義された コンパクト台を持つ (実) M 進 直交 スケール関数・ウェー
ブレット系 の完全な記述（具体的な構成法）が，I. Daubechies [1], P. N. Heller [3],中岡 明 等によって与えられている．

1.3. ウェーブレットの正則性.
次に問題になるのが，構成された 直交 ウェーブレット の性質 である．その１つとして 正則性 (滑らかさ) の評価 が上げられる．ウェーブレット系 ψ i (x) は ϕ(M x − k)の一次結合として表されるので，この問題に関しては，スケール関数の正則性を考察すれば十分である．P. N. Heller – R. O. Wells, Jr. [4] 等は，Sobolev 指数 s2 による Hölder 指数 の評価を用い，2 進・3 進 の場合に，包括的に 直交スケール関数の滑らかさの数値的評価を与えている．我々は，s1 指数 を用いて彼等の評価の一部を改良した [6]．次節以後は，この話題の解説である．直交 ウェーブレット の正則性は，台の幅 や vanishing moments 条件 を上げただけでは，効果的に改善されず，零点条件といった他の条件を加味する必要がある．
1.4. 時間-周波数解析.
以上は，数学的な観点からの考察であったが，工学への応用では，Fourier 解析・
ウェーブレット解析 は 時間-周波数解析 と解釈される．入力信号の時系列は，時間t を変数とする R 上の関数 f (t) として表される．時間-周波数解析では，入力信号
f (t) の中に どの周波数成分が どれだけ含まれるか を 各時刻の周りで 適当な時間幅で検出しようとする．高周波成分は 短い時間幅で検出できるが，中・低周波成分の検出には，その波長に応じた長さの時間幅が必要となる．Fourier 級数展開 は 一定
時間幅での 正弦波 への分解であり，また，Fourier 変換 は (−∞, ∞) 上での積分である．従って，各時刻の周りでの周波数解析のためには，f (t) を局所化することが必要となる．
コンパクト台を持つウェーブレット ψ(t) を用いた ウェーブレット解析では，ウェーブレット に対する dilation 変換 は 周波数変換 及び それに応じた時間幅変換，translation 変換は，中心時刻変換 と見なされ，これにより得られる 直交基底 ψj,k(j, k ∈ Z) や ウェーブレット変換 Wψ は 時間-周波数解析 に適したものとなる．すでに，工学の諸分野において，ウェーブレット解析は，時間-周波数解析 の主要な手段となっている．

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TATSUHIKO YAGASAKI

2. rank M スケール関数 に関する基本事項
本節では rank M スケール関数に関する基本事項を説明し，本論説で考察する問
題を具体的に設定する．以下，スケール関数，ウェーブレット関数 等は，すべて実関数の範囲で考える．また，Fourier 変換は (符号を変えて) 次式で定義する：
Z ∞ˆ
f (ξ) =
f (x)eiξx dx.
−∞

定義 2.1. 関数 ϕ(x) ∈ L2 (R) が次の条件を満たすとき，rank M スケール関数 であ
るという：
X
(∗)
ak = M を満たす実数列 {ak }k∈Z が存在して 次が成り立つ：kϕ(x) =

X

ak ϕ(M x − k).

k∈Z

この関係式をスケール関係，また，数列 {ak }k を スケール列 と呼ぶ．スケールi関係は，ϕ(x) の x = M n−1(i ∈ Z) での値から x = Mi n (i ∈ Z) での値が定まることを意味しており，スケール列 {ak }k が求まれば，関数 ϕ(x) を数値的に求める（グラフを数値的に描く）ことは容易となる．
ϕ(x) の構成という観点からは，ϕ(x) 自身よりも Fourier 変換 ϕ̂(ξ) を扱った方が良い．この際，スケール列に対応するものとして，Fourier 級数

A(ξ) :=

1 X
ak eikξMkが重要になる．これを ϕ(x) のシンボルと呼ぶ．これを用いれば，スケール関係は，次式と同値になる：

ϕ̂(ξ) = A(ξ/M )ϕ̂(ξ/M ).
ϕ(x) がコンパクト台を持つときには，さらに 次式が成り立つ：
ϕ̂(ξ) = ϕ̂(0)∞YA(ξ/M j ).

j=1

スケール関数に関係する概念として，最大次数 N ，長さ L，及び，直交性 がある：
(i) ある三角多項式 Q(ξ) を用いて，シンボル A(ξ) が次の様に表記されるとき，スケール関数 ϕ(x) は 次数 N を持つという：µ¶N
1 + eiξ + · · · + ei(M −1)ξA(ξ) =Q(ξ).
M

ϕ(x) が 次数 N を持つが，N + 1 は持たないとき，ϕ(x) は 最大次数 N を持つ という．最大次数 N における関数 Q(ξ) を，ϕ(x) の既約シンボルと呼ぶ．

(ii) ϕ(x) の長さ L を次式で定める：
L = k1 − k0 + 1(k1 = max{k ∈ Z | ak 6= 0}, k0 = min{k ∈ Z | ak 6= 0}).

ϕ(x) がコンパクト台を持つことは，長さ有限 (L < ∞) と同値になる．

SMOOTHNESS OF SCALING FUNCTIONS AND TRANSFER OPERATORS

83(iii) ϕ(x) が直交であるとは，関数列 {ϕ(x − k)}k∈Z が L2 (R) で直交系を成すことである．このとき，シンボル A(ξ) は，次の “直交条件” を満たす：
M−1
X|A(ξ + 2kπ/M )|2 = 1.

k=1

逆に，シンボル A(ξ) が 直交条件 を満たすとき，長さ有限のスケール関数

ϕ(x) が直交であるための必要十分条件は，Q(ξ) が Cohen 条件 を満たすことである．但し，Q(ξ) が Cohen 条件 を満たす とは，ξ = 0 の近傍を含む Rのコンパクト部分集合 F が存在して 次が成り立つことである：(i) R 上の任意の 2π 周期 非負可測関数 f (ξ) に対して次の等式が成り立つ：ZZf (ξ) dξ =[−π,π]−j(ξ ∈ ∪∞
j=1 M(ii) Q(ξ) 6= 0

f (ξ) dξ ．
FF ).

例えば，Q(ξ) 6= 0 (ξ ∈ [0, π/M ]) ならば，Q(ξ) は Cohen 条件 を満たす．
定義 2.2. 最大次数 N , 長さ L の rank M (実) スケール関数で そのシンボル A(ξ)
が 直交条件 を満たすもの全体のクラスを 記号 ϕM,N,L で表す．L ≥ M N が成り立
つ．特に，L = M N の場合，長さ最小であるといい，このときの ϕM,N,L を ϕM,Nで表す．

ϕM,N,L の完全な記述（具体的な構成法）が [3] によって与えられている.
定理 2.1. (1) 任意の ϕ(x) ∈ ϕM,N,L に対して，R(ξ) = |Q(ξ)|2 は，非負実関数で，次の形をしている：(i) L = M N (長さ最小) の場合：
但し RN (ξ) ≡N−1X

cn (1 − cos ξ)nR(ξ) = RN (ξ)(cn は M , N , n にのみ依存して定まっn=0ているある実定数).
(ii) L = M N + L0 + 1 (L0 ≥ 0) の場合：
X
但し R̃(ξ) = (1 − cos ξ)Nc̃n cos nξR(ξ) = RN (ξ) + R̃(ξ)(L0 6∈ M Z, c̃n ∈ R).

1≤n≤L0n6∈M Z(2) 逆に，(1) の形をした任意の非負実関数 R(ξ) が与えられれば，次の手順で 長さ有限の rank M スケール関数 ϕ(x) で シンボルが直交条件を満たすもの が構成される：
(i) Riesz lemma より ３角多項式 Q(ξ) で |Q(ξ)|2 = R(ξ) を満たすものが取
れる．
´N
³ iξ
i(M −1)ξ
Q(ξ) により A(ξ) が定義され，(ii) A(ξ) = 1+e +···+eM∞Yϕ̂(ξ) = ϕ̂(0)A(ξ/M j ) により ϕ̂(ξ) が定まる．これから，ϕ(x) が定義さ
j=1

れる．

84

TATSUHIKO YAGASAKI

第 1 節 で述べたように，rank M 直交 ウェーブレット の正則性の評価は，付随する直交スケール関数の正則性の評価に帰着する．以下では，次の問題を考察する：
問題 2.1. (1) ϕM,N の正則性の評価(2) 長さ L を最小長 M N から少し伸ばし，既約シンボル Q(ξ) に適当な位置で零点を持たせるとき，ϕM,N,L の正則性は ϕM,N の正則性からどれだけ改善されるか？
問題 (2) において，ϕM,N,L に属するスケール関数で，その既約シンボル Q(ξ) が規定された位置に零点を持つものが存在しない場合もあり，規定された零点が実現されるかどうかは，ϕM,N,L の構成法 (定理 2.1) に基づいて個別に確かめなければならない．|Q(ξ)|2 は cos ξ の多項式であるから，多項式 r(x) で r(cos ξ) = |Q(ξ)|2 を満
たすものが一意に定まる．定理 2.1 (1) に基づく 多項式 r(x) の記述から，実現可能な Q(ξ) の零点の位置に関して次のことが分かる：
0(i) N ≥ 1, ξ1 ∈ ( 3π
5 , π) に対して ϕ ∈ ϕ2,N,2N +4 で r(cos ξ1 ) = r (cos ξ1 ) = 0
を満たすものが存在する．
(ii) M ≥ 3, N ≥ 1 に対して ϕ ∈ ϕM,N,M N +2 で r(cos π) = 0 を満たすものが存在する．
(iii) M ≥ 3, N ≥ 1, ξ1 ∈ [ π2 , π] (M = 3, ξ1 = 2π
3 を除く) に対して ϕ ∈ϕM,N,M N +3 で r(cos ξ1 ) = r0 (cos ξ1 ) = 0 を満たすものが存在する．

3. sp 指数 & Transfer operators
本節では，ϕM,N,L の正則性 を評価する際に必要となる sp 指数 及び transfer
operator の spectral radius についての基本事項を説明する．

3.1. 滑らかさの指標 – Hölder 指数，sp 指数.
関数 f (x) の滑らかさの指標として Hölder 指数 α(f ) を用いることができる．
n ∈ Z≥0 に対して，C n は n 回連続微分可能な関数のクラスを表す．さらに，α = n+σ(σ ∈ (0, 1)) に対して，クラス C α を次で定義する：¯()¯|f (n) (x) − f (n) (y)|αn ¯C := f ∈ C ¯ sup<∞ .
¯ x6=y|x − y|σ関数 f (x) の Hölder 指数 α(f ) は，次式で定義される：

α(f ) := sup{α ≥ 0 | f ∈ C α }.
Sobolev 埋め込み定理 が示唆するように，Hölder 指数は Sobolev 指数で評価される．関数 f ∈ L2 (R) に対して，その Sobolev 指数 s(f ) は 次式で定義される：
¯Z ∞
½
¾
¯
sˆ
2
¯
s(f ) := sup s ∈ R ¯|(1 + |ξ|) f (ξ)| dξ < ∞ .
−∞一般の p > 0 に対しても，f の sp 指数 sp (f ) を次式で定義することができる：
¯Z ∞
¾
½
¯
sˆ
p
¯
|(1 + |ξ|) f (ξ)| dξ < ∞ .
sp (f ) := sup s ∈ R ¯−∞

2

命題 3.1. f ∈ L (R) がコンパクト台を持つ時，次の不等式が成立する ([6, Proposition4.1])：

sp (f ) − p−1
p ≤ s1 (f ) ≤ α(f ) ≤ sr (f )(p ≥ 1, r ≥ 2)

SMOOTHNESS OF SCALING FUNCTIONS AND TRANSFER OPERATORS

85

記号を簡略化するために，ϕM,N,L に属する任意のスケール関数に関して成立する
場合には，sp (ϕM,N,L ), ϕM,N,L ∈ C n 等の記号を用いる．
ϕM,N,L の sp 指数は，既約シンボルに付随する transfer operator の spectral radius
を用いて評価することが出来る．次節では，transfer operator に関する基本事項を説明する．

3.2. Transfer operators & Spectral radius.
Transfer operator は力学系に付随した概念である．ここでは，１次元区間上の折れ線写像 に付随する transfer operator を扱う．
折れ線写像 κ : [0, π]→ [0,¶µ π] を次式で定める：
2i
2i + 1
2i


π
π≤x≤
π
M x −M
M
Mµ¶
κ(x) =
2i + 22i + 12i + 2


π−x
π≤x≤
π.
M
M
M
M
·
¸
·
¸
i
i+1
i
i+1
一次変換 κ :π,π → [0, π] の逆一次変換を θi : [0, π] →
π,
π
M
M
M
M
で表す．
可測関数 q, f に対して，関数 Tq (f ), Uq (f ) を次式で定義する：

Tq (f )(x) :=M−1X

q(θi (x))f (θi (x)),

Uq (f )(x) := q(x)f (κ(x)).

i=1

さらに，Tq j , Uq j (j = 1, 2, · · · ) は，それぞれ Tq , Uq の j 回合成作用素 を表す．Tq j ,Uq j は随伴作用素の関係にある．すなわち q, f , g を [0, π] 上の有界可測関数とする
とき，次式が成り立つ：
Z π
Z π
j
j
Tq (f ) g dx = M
f Uq j (g) dx(j = 1, 2, · · · ).
00

L∞ ([0, π]) は，[0, π] 上の実数値有界可測関数全体の成す線形空間にノルム
kf k := ess sup {|f (x)| : x ∈ [0, π]}を与えた Banach 空間を表す．L∞ ([0, π]) は 次の２つの cone を含んでいる：

K := {f ∈ L∞ ([0, π]) : f ≥ 0 a.e.},

K0 := {f ∈ L∞ ([0, π]) : ess inf[0,π] f > 0}.

q ∈ K に対して，作用素 Tq : L∞ ([0, π]) → L∞ ([0, π]) を q に付随する transfer
operator と呼ぶ．Tq は 有界線形作用素で positive (Tq (K) ⊂ K) である．Tq のspectral radius ρ(Tq ) は 次で定義される：
1

ρ(Tq ) := lim kTq n k n .
n→∞1ρ(Tq ) = lim kTq n (f )k n (f ∈ K0 ) が成り立ち，r, q ∈ K, r ≤ q (a.e.) ならばn→∞ρ(Tr ) ≤ ρ(Tq ) ≤ M kqk となる．
C([0, π]) で 連続関数のなす L∞ ([0, π]) の部分空間を表す．C([0, π]) は，次の coneを含んでいるK̃ := K ∩ C([0, π]), K̃0 := K0 ∩ C([0, π]).
q ∈ K̃ のとき，Tq (C([0, π])) ⊂ C([0, π]) となる．ρ(Tq |C([0,π]) ) = ρ(Tq ) なので，簡単のため Tq |C([0,π]) も Tq で表すことにする．もし q ∈ K̃0 ならば，任意の f ∈ K̃,
f 6= 0 に対して ある k ≥ 1 が存在して Tq k (f ) ∈ K̃0 となる．
Tq は，スペクトルについて 次の様な性質を持つ (cf. [6, Section 3])：

86

TATSUHIKO YAGASAKI

定理 3.1. q ∈ K̃0 は Hölder 連続とし，f ∈ K̃0 とする．fn := Tq n (f ), gn :=(n ≥ 1) と置く．
このとき，次が成り立つ：(i) 次を満たす g ∈ K̃ が一意に存在する： kgk = 1,
Tq (g) = λg.
このとき，さらに，g ∈ K̃0 , λ = ρ(Tq ) が成り立つ．
(ii) gn は g に一様収束する．
fn+1
(iii)
は ρ(Tq ) に一様収束する．
fn
fn+1
fn+1
(iv) min% ρ(Tq ), max& ρ(Tq ).
[0,π] fn[0,π] fn

fn
kfn k

λ > 0 が存在して命題 3.2. α, β を正定数とする．p, q ∈ K0 で α ≤ p, q ≤ β ならば 次式 が成立する：|ρ(Tp ) − ρ(Tq )| ≤ M

αkp − qk.
β

4. ϕM,N,L の正則性の評価
以下，ϕ(x) ∈ ϕM,N,L とし， Q(ξ) を ϕ(x) の既約シンボルとする．

4.1. s2 指数の評価. — [2, 4]
T. Eirola [2], P. N. Heller – R. O. Wells, Jr. [4] 等は，q(ξ) = |Q(ξ)|2 を weight 関
数とする transfer operator T|Q|2 を用いて，s2 (ϕ) の 次の様な表示式を与えた：
定理 4.1.

s2 (ϕ) = N −1
logM ρ(T|Q|2 ).
2|Q|2 は cos ξ の多項式になり，作用素 T|Q|2 は cos kξ で張られる 標準的な 有限次
元不変部分空間 を持つ．ρ(T|Q|2 ) は T|Q|2 の この部分空間への制限を表す positive
matrix の 最大固有値 に一致する．このことは，T|Q|2 が正値固有関数を持つ事を意
味し，上の表示式の証明で重要であると同時に，s2 (ϕ) が行列の固有値の数値計算で容易に求められることも意味する．
s2 (ϕ) −1≤ α(ϕ) ≤ s2 (ϕ)
2であるから，これにより，ϕ の正則性が 数値的に評価されることになる．彼等は，
M = 2, 3 の場合に，この数値計算を行い，

ϕ2,7 ∈ C 2 ,ϕ2,11 ∈ C 3 ,ϕ3,9 ∈ C 1

といった結果や，s2 (ϕM,N ) の 上・下からの評価，さらに，N → ∞ のときの漸近的な振る舞い 等 について 包括的に調べている．
長さ最小の場合 |Q| > 0 であるが，長さ L を 最小長 M N から少し伸ばせば，|Q| が適当な位置に零点を持ち，max |Q| が小さくなるような スケール関数 ϕ(x) ∈ ϕM,N,Lを構成することが出来る．彼等は，Q(ξ) が 写像 ξ 7→ M ξ (mod 2π) の周期点や準周期点に零点を持つと，最小長の場合に比べて正則性が著しく改善することを示している．

SMOOTHNESS OF SCALING FUNCTIONS AND TRANSFER OPERATORS

87

4.2. sp 指数の評価. — [6, Section 4]
一般の p > 0 の場合，weight 関数として q(ξ) := |Q(ξ)|p をとると，q(ξ) は 2π 周期を持つ Hölder 連続な偶関数となり，特に，ϕ(x) が長さ最小の場合には，q(ξ) は正値 C ∞ 関数で [0, π] 上で 狭義 単調増加である．現時点では，Q(ξ) に関する適当な仮定無しで，常に Tq が正固有関数を持つかどうかは不明である．このため，以下の 定理 4.2 の評価式は不等式の形になっている．
任意の f ∈ K0 に対して Tq (f ) ∈ K0 となり，µ(f ), λ(f ) > 0 をµ(f ) := ess inf[0,π]Tq (f ),fλ(f ) := ess sup[0,π]Tq (f )fで定義することができる．µ(f ) ≤ ρ(Tq ) ≤ λ(f ) が成り立つ．
定理 4.2. 次の不等式が成り立つ：
1logM ρ(Tq ).
(i) sp (ϕ) ≥ N −p(ii) Q が Cohen 条件を満たすとき，sp (ϕ) ≤ N −K0 } ≤ N .
系 4.1. Q(ξ) が [0, π] に零点を持たないとき，

1sup{logM µ(f ) | f ∈psp (ϕ) = N −1
logM ρ(Tq ).
p

4.3. sp 指数の数値計算 — 階段関数による近似. — [6, Section 5]
1
s2 (ϕ) − ≤ s1 (ϕ) ≤ α(ϕ) であるから，第 4.1 節で s2 (ϕ) から得られている ϕ2の正則性の評価を，s1 (ϕ) の数値計算により さらに改良できる可能性がある．ρ(Tq )を数値計算で数値的に評価するためには，離散化による有限近似が必要となる．p = 2の場合は，Tq が，標準的な有限次元不変部分空間を持ち，ρ(Tq ) の数値計算は，この部分空間に制限された Tq を表す行列の固有値の計算に帰着した．しかし，一般のp の場合には，このような 標準的な 有限次元不変部分空間 は見当たらない．そこ
で，自然に考えられる方法は，q を階段関数で近似する方法である．すなわち，q を階段関数 q ± を用いて q − ≤ q ≤ q + という形に 挟んで ρ(Tq ) を ρ(Tq± ) で評価するわけである．
N ≥ 1 に対して，SN ([0, π]) で，[0, π] を N 等分し，その各小区間上定数となる階段関数全体の成す L∞ ([0, π]) の N 次元部分空間 を表す．m ≥ 1 とする．
q ∈ SM 2 m ([0, π]), q ≥ 0 のとき，Tq (SM m ([0, π])) ⊂ SM m ([0, π]) となる．q − , q + ∈SM 2 m ([0, π]) を，各々 q − ≤ q ≤ q + a.e. を満たす 最大・最小階段関数とする．
f ∈ SM m ([0, π]) とする．fn± := Tq±n (f ) ∈ SM m ([0, π]) (n = 1, 2, · · · ) と置く．m が±十分大きければ fn± ∈ K0 となり µ±n , λn を次の様に定義することができる：µ±
n := ess inf[0,π]

±
fn+1
,
fn±

λ±
n := ess sup[0,π]

±
fn+1
fn±

(n = 1, 2, · · · ).

±±±次の関係が成り立つ：µ±n ≤ µn+1 ≤ ρ(Tq ± ) ≤ λn+1 ≤ λn .
±さらに q ± ∈ K0 のときには，次が成り立つ：µ±n % ρ(Tq ± ), λn & ρ(Tq ± ) (n → ∞).
Theorem 4.1 より，次が成り立つ：

命題 4.1. (1) m が十分大きいとき，次の不等式が成り立つ：
1(i) sp (ϕ) ≥ N −inf logM λ+n.
p n1(ii) Q が Cohen 条件を満たすとき，sp (ϕ) ≤ N −sup logM µ−
n.
p n

88

TATSUHIKO YAGASAKI(2) Q(ξ) が [0, π] に零点を持たないとき，次が成り立つ：
inf m,n λ+n.

ρ(Tq ) = supm,n µ−n =4.4. s1 指数の数値計算の結果. — [6, Section 6]
第 4.3 節の階段関数近似の議論に基づいて s1 指数の数値計算を行った．その結果と 第 4.1 節の結果を組み合わせることで導かれる ϕM,N,L の正則性に関する評価の中で，特に，顕著な結果を以下にリストする．(r(x) は |Q(ξ)|2 = r(cos ξ) を満たす多項式であった.)(i) ϕ2,62 ∈ C 2 , ϕ2,8 6∈ C 2 , ϕ2,9 ∈ C 3 , ϕ2,12 6∈ C 4 , ϕ2,13 ∈ C 4 ;(ii) ϕ3,4 ∈ C 1 , ϕ3,77 6∈ C 2 , ϕ3,78 ∈ C 2 ;(iii) ϕ2,6,16 ∈ C 3
ϕ2,9,22 ∈ C 4
ϕ2,13,30 ∈ C 5

5π
if r(cos ξ) = r0 (cos ξ) = 0 for ξ = 4π
5 or 6 ;3π
0if r(cos ξ) = r (cos ξ) = 0 for ξ = 4 ;3πif r(cos ξ) = r0 (cos ξ) = 0 for ξ = 2π
3 or 4 ;(iv) ϕ3,3,11 ∈ C 1 , ϕ3,6,20 ∈ C 2 , ϕ3,9,29 ∈ C 3 , ϕ3,13,41 ∈ C 4
1(v) ϕ3,2,9 ∈ C
ϕ3,5,18 ∈ C 2
ϕ3,8,27 ∈ C 3
ϕ3,11,36 ∈ C 4

if
if
if
if

if r(cos π) = 0;

ξ = 5π6 ;5π
ξ = 6 or π;0r(cos ξ) = r (cos ξ) = 0 forr(cos ξ) = r0 (cos ξ) = 0 forr(cos ξ) = r0 (cos ξ) = 0 for ξ = π;r(cos ξ) = r0 (cos ξ) = 0 for ξ = π;

最後に，s2 (ϕM,N ) − 1/2 と s1 (ϕM,N ) を比較する数値計算の結果をリストする：

M =2
N
1
2
3
4
5
6
7
8
9
10
11
12
13

M =3

s2 (ϕ2,N ) − 1/2 s1 (ϕ2,N )000.50000.521
0.91500.980
1.27561.392
1.59681.768
1.88842.117
2.15872.442
2.4147
2.747
2.66173.036
2.9027
3.310
3.13983.572
3.37403.826
3.6060
4.072

N
1
2
3
4
5
6
7
8
9
10
11
12
13
77
78

s2 (ϕ3,N ) − 1/2 s1 (ϕ3,N )000.40870.443
0.65990.779
0.79501.031
0.86651.211
0.91331.331
0.94991.410
0.98091.462
1.0081
1.499
1.0323
1.528
1.0542
1.552
1.0741
1.573
1.09251.592
1.49991.999
1.5016
2.002

5. 最後にp が一般の場合，transfer operator Tq の性質は，まだ，十分に理解されたとはいえない．今後，力学系との関連の中で，Tq の spectral radius の性質を明確にしたいと考えている．

SMOOTHNESS OF SCALING FUNCTIONS AND TRANSFER OPERATORS

89

References[1] I. Daubechies, Ten Lectures on Wavelets, CBMS Conf. Ser., Vol. 61, SIAM, Philadelphia,
1992.
[2] T. Eirola, Sobolev characerization of solutions of dilation equvations, SIAM J. Math. Anal.
23 (1992) 1015–1030.
[3] P. N. Heller, Rank M wavelet matrices with N vanishing moments, SIAM J. Math. Anal.
Appl. 16 (1995) 502–518.
[4] P. N. Heller and R. O. Wells, Jr., Sobolev regularity for rank M wavelets, Technical Report
96-08, Computational Mathmatics Laboratory, Rice University, 1996.
[5] C. K. Chui, An Introduction to Wavelets, Academic Press Inc., Orlando, 1992.
[6] F. Maitani, A. Nakaoka, H. Okura and T. Yagasaki, Smoothness of rank M scaling functions,Applied and Computational Harmonic Analysis, 13 (2002) 116–137.
Department of Mathematics, Kyoto Institute of Technology, Matsugasaki, Sakyoku,
Kyoto 606, Japan
E-mail address: yagasaki@ipc.kit.ac.jp

Proceedings of General Topology Symposium, p. 91 - p. 100, 2002

http://gt2002.h.kobe-u.ac.jp/

CLOSED IMAGES OF SPACES HAVING g-FUNCTIONSIWAO YOSHIOKA

1. Introduction and Definitions
In §2, we consider closed images of ks-spaces [34] (ks-spaces are equivalent to
k-semistratifiable spaces in the realm of T2 -spaces). Lutzer [22] showed that theclosed image of a paracompact k-semistratifiable space (indeed, the image of a ksemistratifiable space by a closed, compact-covering map) is k-semistratifiable. We
prove that the closed image of a ks, Fre̋chet T1 -space or ks, q, regular space is ks
and the finite-to-one closed image of a N agata space is N agata. We also provethat every M3 , q-space is N agata.
In §3, we define the class of weak contraconvergent (=wcc) spaces which containthe class of M CP spaces [10] or contraconvergent spaces [31] and are contained in
the class of β-spaces. And we prove that the closed images or the pre-images byquasi-perfect maps of wcc-spaces are wcc-spaces. Also, we prove that the class of
wcc- and q-spaces are equivalent to the class of wN -spaces. Moreover, we prove thatevery fiber of a closed map from a wcc-space onto a q-space is countably compact.
In §4, we introduce the concept of strongly α-spaces which contain the classof paracompact spaces with Gδ -diagonals and are contained in the class of αspaces. And we prove that every strongly α, wcc-space is k-semistratifiable andevery strongly α, wcc, wθ-space is metrizable.
In §5, it is showed that quasi-perfect images of γ- (wγ-) spaces are γ (wγ) andopen closed images of γ- (wγ-) spaces are γ (wγ).
Throughout this paper, all maps are onto and we assume no separation axioms
unless otherwise stated. The set of natural numbers is denoted by N. Finally, we
refer the reader to [6] for undefined terms.
Definition 1.1. For a space X, a structure ({gn (x)} | x ∈ X) is called a gstructure if gn (x) is an open neighbourhood of x and gn+1 (x) ⊂ gn (x) for any
x ∈ X and every n ∈ N. For a subset A of X, we put gn (A) = ∪{gn (x) | x ∈ A}.
We now consider the following conditions on a g-structure G = ({gn (x)} | x ∈ X)
of a space X.
(A) If gn (x) ∩ gn (xn ) 6= ∅(n ≥ 1), then x is a cluster point of {xn }n .
(B) If gn (x) ∩ gn (xn ) 6= ∅(n ≥ 1), then {xn }n has a cluster point.
(C) If x ∈ gn (xn )(n ≥ 1), then {xn }n −→ x and if H is closed in X, then∩n≥1 gn (H) = H.
(D) If yn ∈ gn (xn )(n ≥ 1) and y is a cluster point of {yn }n , then y is a cluster
point of {xn }n .
(E) If yn ∈ gn (xn )(n ≥ 1) and {yn }n −→ y, then {xn }n −→ y.
(F ) If x ∈ gn (xn )(n ≥ 1), then {xn }n −→ x.
91

92

IWAO YOSHIOKA(G) If x ∈ gn (xn )(n ≥ 1), then {xn }n has a cluster point.
(H) {x} = ∩n≥1 gn (x) and, gn (y) ⊂ gn (x) if y ∈ gn (x).
(I) If yn ∈ gn (p), xn ∈ gn (yn )(n ≥ 1), then p is a cluster point of {xn }n .
(J) If yn ∈ gn (p), xn ∈ gn (yn )(n ≥ 1), then {xn }n has a cluster point.
(K) If yn ∈ gn (p), xn , p ∈ gn (yn )(n ≥ 1), then p is a cluster point of {xn }n .
(L) If yn ∈ gn (p), xn , p ∈ gn (yn )(n ≥ 1), then {xn }n has a cluster point.
(M ) If xn ∈ gn (x)(n ≥ 1), then {xn }n has a cluster point.
A T1 -space satisfying (A) ((B)) is called a N agata space [2, 14] (a wN -space[17]) and G is called a N agata structure (a wN -structure, respectively). A T1 -spacesatisfying (C) is called a stratif iable space [1, 12] and G is called a stratifiable
structure.
It is well known that [11] a space is stratifiable if and only if it is M2 [2] and [2]the closed image of a stratifiable space is stratifiable.
A space satisfying (D) ((E)) is called a contraconvergent space [31] (a ks-space[34], which was called a strongly-quasi-N agata space in [18]) and G is called acontraconvergent structure (a ks-structure, respectively). A space satisfying (F ) is
called a semistratif iable space [5] and G is called a semistratifiable structure.
Hodel [16] called a β-space (an α-space) for a space satisfying (G) ((H), respectively) and proved that a T2 -space is semistratifiable if and only if it is a αand β-space. It is known that every closed image or finite-to-one open image of asemistratifiable T2 -space is semistratifiable [9: Theorem 2.1].
Moreover, a space satisfying (I) ((J)) is called a γ-space [17](a wγ-space [17])and G is called a γ-structure (a wγ-structure, respectively).
A space satisfying (K) ((L)) is called a θ-space [17] (a wθ-space [17]) and G iscalled a θ-structure (a wθ-structure, respectively). Finally, a space satisfying (M )is called a q-space [25] and G is called a q-structure.
Every wθ-space or wN -space is a q-space.
We define a k-semistratifiable space [22] by a equivalent condition [8: Theorem
3] which is true for no separation axiom.
Definition 1.2. A space X is a k-semistratif iable if X has a g-structure({gn (x)} | x ∈ X) such that if K ∩ F = ∅, where K is compact and F is closed, thenK ∩ gm (F ) = ∅ for some m ∈ N.
Proposition 1.3 [34: Proposition 3]. The following implications hold.
Nagata spaces =⇒ stratifiable spaces =⇒ contraconvergent spaces=⇒ k-semistratifiable spaces =⇒ ks-spaces =⇒ semistratifiable spaces.
Every ks-space is a σ-space [15] and the closed image of a regular σ-space is σ[12: Corollary 4.12], where a space with a σ-locally finite network is called a σ-space[30].
The next result follows from [3: Corollary 3.A.1].
Proposition 1.4. Every countably compact ks (or γ), T2 -space is compact
metrizable.

CLOSED IMAGES OF SPACES HAVING g-FUNCTIONS

93

Gao [8: Theorem 5] showed that every ks, T2 -space is k-semistratifiable. Here,
we give a simpler proof.
Proposition 1.5. Every ks, T2 -space is k-semistratifiable.
Proposition 1.6. Every quasi-perfect map defined on a ks (or γ), T2 - space isperfect.

2. Contraconvergent spaces
With a view to studying closed images of k-semistratifiable spaces, we first consider the closed images of contraconvergent spaces.
Let f be a map from a space X to a space Y and A be a subset of X. Then byA∗ , we denote the subset ∪{V : open in Y | f −1 (V ) ⊂ A} of Y .
Theorem 2.1. Let f : X −→ Y be a closed map. If X is a contraconvergent
T1 -space, then Y is contraconvergent.
Under no separation axiom, we have the following result.
Theorem 2.2. Let f : X −→ Y be a quasi-perfect map. If X is a contraconvergent space, then Y is contraconvergent.
The following example shows that all spaces given in Proposition 1.3, γ-spacesand θ-spaces are not necessarily inverse invariant by perfect maps.
Example 2.3. Let X = N×βN (N has the discrete topology ) and p : X −→ N
be a projection. Then, p is a perfect map and N is completely metrizable. But, it
is easily seen that X is a wN, wγ-space which is not first countable (thus X is notθ) and not semistratifiable (thus X is not α).
Mizokami and Shimane [28] showed that every k, M3 -space is M1 . But, thespace Y in [26: Example 10.1] is an M1 , Fre̋chet space which is not N agata. The
following theorem shows that every M3 , q-space is N agata. The conditions forwN -spaces to be N agata are studied in [19].
Theorem 2.4. For a T1 -space X, the following conditions are equivalent.
(1) X is a Nagata space.
(2) X is an M1 , q-space.
(3) X is a contraconvergent q-space.
(4) X is a ks, q, regular space.
Theorem 2.5. Every ks, Fre̋chet-space is contraconvergent.
Lutzer [22: Example 4.3] describes that the perfect image of a N agata space is
not even a q-space. On the other hand, every wN -space is preserved by a finite-toone closed map [10: Proposition 18]. Therefore by Theorems 2.1, 2.4, we have the

94

IWAO YOSHIOKA

following result.
Theorem 2.6. The finite-to-one closed image of a Nagata space is Nagata.
Remark 2.7. That the finite-to-one closed image of a q-space is q can be directly
proved by analogy to the proof of [10: Proposition 18].
By Proposition 1.6 and [22: Proposition 2.5], the quasi-perfect image of a ksemistratifiable T2 -space is k-semistratifiable.
I do not know whether the closed (even perfect) image of a ks, T1 -spaces is ks
or not. In [29: Theorem 3.3], Mohamad asserts that the closed image of a regular
ks-space is ks. But to me, his proof is not clear.
For closed images of ks-spaces, we have the following results by Theorems 2.1,
2.2, 2.4 and 2.5.
Theorem 2.8. (1) Let f : X −→ Y be a closed map. If X is a ks, Fre̋chetT1 -space or ks, q, regular space, then Y is contraconvergent.
(2) Let f : X −→ Y be a quasi-perfect map. If X is a ks, Fre̋chet space, then Y
is contraconvergent.

3. Weak contraconvergent spaces
For convenience, we introduce the following notation: if (An )n≥1 and (Bn )n≥1
are two sequences of subsets, we write (An ) ¹ (Bn ) if An ⊂ Bn for each n ∈ N.
Definition 3.1 [10]. A space X is said to be monotonically countably paracopmact (=M CP ) if there exists an operator U assigning to each decreasing sequence (Dj )j≥1 of closed subsets with empty intersection, a sequence of open subsets
U ((Dj )) = (U (n, (Dj )))n≥1 such that
(1) Dn ⊂ U (n, (Dj )) for each n ∈ N,(2) ∩n≥1 U (n, (Dj )) = ∅,(3) if (Dn ) ¹ (En ), then U ((Dj )) ¹ U ((Ej )).
Clearly, every M CP space is countably paracompact.
Definition 3.2. A space X is said to be weak contraconvergent(=wcc) if there
exists a g-structure of X such that if yn ∈ gn (xn )(n ≥ 1) and {yn }n has a cluster
point, then {xn }n has a cluster point.
It is clear that every contraconvergent space is wcc.
Theorem 3.3. The following implications hold for a T1 -space X.
(1) a wN-space=⇒(2) an MCP space =⇒ (3) a wcc-space =⇒ (4) a β-space.
Example 3.4. (1) There exists a Moore (hence, semistratifiable [17]) space
which is neither wcc nor ks.
(2) There exists a wcc-space which is not semistratifiable.

CLOSED IMAGES OF SPACES HAVING g-FUNCTIONS

95(3) There exists a wcc-space which is not countably paracompact, hence not
M CP .
In [10], Good, Knight and Stares showed the results that a space X is wN if
and only if it is M CP, q and, X is metrizable if and only if it is M CP , Moore or
M CP, γ.
Theorem 3.5. X is a wN-space if, and only if, it is a wcc, q-space.
As to metrizations of wcc-spaces, we have the following results which can weaken
wcc-spaces to quasi-N agata spaces [20; 24; 34].
Corollary 3.6. A T2 -space X is metrizable if it satisfies any one of the following
conditions. (1) X is a wcc, developable space.
(2) X is a wcc, γ-space.
Remark 3.7. [0, ω1 ) with the order topology is a wcc, θ-space which is not
metrizable [17: Example 4.12].
In [10: Example 15], it is described that there exists an M CP space which is
not preserved by a closed map. On the other hand, weak contraconvergentness ispreserved by a closed map. The following two theorems are proved by analogy tothe proofs of Theorems 2.1 and 2.2.
Theorem 3.8. Let f : X −→ Y be a closed map. If X is a wcc, T1 -space, thenY is wcc.
Thorem 3.9. Let f : X −→ Y be a quasi-perfect map. If X is a wcc-space, thenY is wcc.
The following example asserts that ks-spaces or wcc-spaces are not necessarily
preserved by finite-to-one open, compact-covering maps.
Example 3.10. Michael [27: Example 9.1] gave the finite-to-one open, compactcovering map from a completely metrizable space X to a metacompact, locally completely metrizable, non-metrizable Tychonoff space Y which is not C̆ech-complete.
Then Y is a semistratifiable γ, Moore space [9: Theorems 2.1, 4.1; 17: Corollary4.6]. But if Y is ks or wcc, then Y is metrizable by [34: Theorem 3] or Corollary
3.6. This is a contradiction.
Note that every locally metrizable, wcc- (or ks-) space is metrizable by [9], [34]or Corollary 3.6.
Also, from [4: Example 6.6], one can see that ks-spaces or wcc-spaces are notnecessarily preserved by two-to-one open maps.
Theorem 3.11. Let f : X −→ Y be a quasi-perfect map. If Y is a wcc-space,then X is a wcc-space.

96

IWAO YOSHIOKA

Corollary 3.12. Let f : X −→ Y be a quasi-perfect map. If X is a γ, T2 -spaceand Y is a wcc-space, then both X and Y are metrizable.
Corollary 3.13 [10]. Letf : X −→ Y be a quasi-perfect map. If Y is a β-space,then X is β-space.
Note that [32] the closed image of a β, T1 -space is β, which is also proved inCase 1 of Theorem 2.1.
Corollary 3.14. Let f : X −→ Y be a quasi-perfect map. If Y is a q-space,then X is a q-space.
Although the perfect pre-imsge of a N agata space is not necessarily N agata by
Example 2.3, the following theorem follows from Theorems 3.5, 3.11 and Corollary
3.14.
Theorem 3.15. Let f : X −→ Y be a quasi-perfect map. If Y is a wN -space,then X is a wN -space.
The first statement of the following corollary makes slightly better [33: Theorem
12; 32: Theorem 3]
Corollary 3.16. Let f be a closed map from a T1 -space X onto a q-space Y.
Then the following statements hold.
(1) If X is a wcc-space, then Y is wN.
(2) If X is a contraconvergent space, then Y is Nagata.
Let f be a closed map from a space X onto a q-space Y . Then, it is well-known
that ∂f −1 (y) is countably compact if X is normal [26] or countably paracompact[32]. Although wcc-spaces are not necessarily countably paracompact, the similar
result follows.
Theorem 3.17. Let f : X −→ Y be a closed map. If X is a wcc, T1 -space andY is a q-space, then ∂f −1 (y) is countably compact for any y ∈ Y .
Moreover if X is semistratifiable T2 , then ∂f −1 (y) is compact.
Lutzer [22: Example 4.3] showed that there exists the perfect map from the
N agata space which is not of countable type to the space which is not q. In
Theorem 3.17 we consider the conditions for Y to be a q-space when ∂f −1 (y) is
compact. For that, we present the following two properties of a space X.
(a) X is a wcc, semistratifiable T1 -space of countable type, where X is of countable type if every compact subset of X has a countable character.
(b) X is a N agata space of countable type.
Condition (a) is strictly weaker than (b) since the space X in the below Example
4.9 satisfies (a) but it is not ks.
Corollary 3.18. Suppose that X satisfies (a) or (b). Then for a closed map
f : X −→ Y , the following conditions are equivalent.

CLOSED IMAGES OF SPACES HAVING g-FUNCTIONS

97(1) Y satisfies (a) or (b), respectively.
(2) Y is a q-space.
(3) ∂f −1 (y) is compact for any y ∈ Y .

4. Strongly α-structures
Definition 4.1. A space X is called strongly α if for any x ∈ X and each n ∈ N,
there exists an open neighbourhood gn (x) of x such that(a) ∩n≥1 gn (x) = {x} and,(b) gn (y) ⊂ gn (x) if y ∈ gn (x).
Here, we can assume that the sequence {gn (x)} is decreasing.
Evidently, every strongly α-space is a T2 -space.
Let consider the following properties (P1 ) and (P2 ) of a space.
(P1 ) There exists a sequence {Fn } consisting of closure-preserving closed covers
of a space X such that, if x 6= y then there exists m ∈ N such that for any
p ∈ X, x ∈/ Fp or y ∈/ Fp for some Fp ∈ Fm with p ∈ Fp .
(P2 ) There exists a sequence {Un } consisting of point-finite open covers of aspace X such that, if x 6= y then there exists m ∈ N such that for any
p ∈ X, x ∈ V or y ∈ V for some V ∈ Um with p ∈/ V.
Definition 4.2. A space X is said to have a strong Gδ -diagonal if X has
a sequence {Gn } of open covers such that whenever x 6= y, there exists m ∈ N
satisfying that x ∈/ st(p, Gm ) or y ∈/ st(p, Gm ) for any p ∈ X. The sequence {Gn } iscalled a strong Gδ -diagonal sequence.
Definition 4.3. A space X is called subparacompact(metacompact) if every
open cover of X has a σ-discrete closed refinement (a point finite open refinement).
Every semistratifiable space is subparacompact [5; 12] and every subparacompact
space with Gδ -diagonal is α [16].
In the realm of paracompact T2 -spaces, the existence of a strong Gδ -diagonal isequivalent to it of a Gδ -diagonal.
Proposition 4.4. (1) If X satisfies (P1 ), then X is strongly α.
(2) If X satisfies (P2 ), then X is strongly α.
(3) If X is a subparacompact space with a strong Gδ -diagonal, then X satisfies(P1 ).
(4) If X is a metacompact space with a strong Gδ -diagonal, then X satisfies (P2 ).
(5) A submetrizable space X satisfies (P1 ) and (P2 ) and hence, every paracompact T2 -space with a Gδ -diagonl also satisfies (P1 ) and (P2 ).
Theorem 4.5. Every developable T2 -space X is a strongly α-space with a strongGδ -diagonal.
Note that every stratifiable space or Sorgenfrey line is strongly α, because it isa paracompact T2 -space with a Gδ -diagonal.

98

IWAO YOSHIOKA

Theorm 4.6. Let f : X −→ Y be a finite-to-one open closed map. If X is astrongly α-space, then Y is strongly α.
Proposition 4.7. For a strongly α-space X, the following statements hold.
(1) If X is a wcc-space, then X is k-semistratifiable.
(2) If X is a wγ-space, then X is γ.
(3) If X is a wθ-space, then X is θ.
Theorem 4.8. For a space X, the following conditions are equivalent.
(1) X is a metrizable space.
(2) X is a strongly α, wM -space.
(3) X is a strongly α, wcc, wθ-space.
(4) X is a strongly α, ks, wθ-space.
Example 4.9. There exists a compact γ, α (hence, semistratifiable), T1 -space
X which is neither ks nor strongly α. Moreover, X is of countable type.
Let X be a space X = (N, O) with the topology O = {G ⊂ N ||N \ G| < ω0 }.
Then X is a compact T1 -space which is not T2 . To see that X is α, let An ={k | k ≥ n} for each n ∈ N. For each x ∈ X and each n ∈ N, let gn (x) = {x} ∪ An .
Then G = ({gn (x)} | x ∈ X) is an α-structure. Therefore X is semistratifiable [16:Theorem 5.2]. We show that G is a γ-structure. Let yn ∈ gn (p), xn ∈ gn (yn ) for
each n ∈ N. Since G is an α-structure, xn ∈ gn (p)(n ≥ 1). If p ∈/ {xn | n ≥ m}for some m ∈ N, then xn ∈ An (n ≥ m) and {xn }n≥m is infinite. Hence {xn }nconverges to p. Next, if X is ks, then X is N agata [34: Theorem 2] and if X isstrongly α, then X is T2 , which are contradictions. Last, we prove that X is ofcountable type. For any compact subset K of X, let A = {X \ F | F is any finite
subset of X \ K}. Then A is a countable base of K.

5. γ-spaces
The irreducible closed images of γ-spaces are not necesarily γ (even q) [26: Example 10.1] (or [6: Problem 5.5.12]). However, Gittings [9: Theorem 4.1] showedthat a finite-to-one open image of a γ-space (a wγ-space) also is a γ-space (a wγspace, respectively). In this section, we study open closed images or quasi-perfectimages of γ-spaces or wγ-spaces. For that reason, we need the following lemma.
Lemma 5.1. Let G = ({gn (x)} | x ∈ X) be a g-structure of a space X. Then thefollowing statements hold.
(1) G is a γ-structure if and only if the sequence {yn } has a cluster point x
whenever yn ∈ gn (xn )(n ≥ 1) and the sequence {xn } has a cluster point x [21] if
and only if the sequence {yn } −→ x whenever yn ∈ gn (xn )(n ≥ 1) and the sequence{xn } −→ x [34].
(2) G is a wγ-strucyure if and only if the sequence {yn } has a cluster point
whenever yn ∈ gn (xn )(n ≥ 1) and the sequence {xn } has a cluster point.
Theorem 5.2. Let f : X −→ Y be a quasi-perfect map. Then the following
statements hold.

CLOSED IMAGES OF SPACES HAVING g-FUNCTIONS

99(1) If X is a γ-space, then Y is γ.
(2) If X is a wγ-space, then Y is wγ.
Example 5.3. θ-spaces are not necessarily preserved by quasi-perfect maps.
Although Example 2.3 asserts that the perfect pre-image of a γ-space is notnecessarily γ, the quasi-perfect pre-images of wγ-spaces also are wγ.
Theorem 5.4. Let f : X −→ Y be a quasi-perfect map. If Y is a wγ-space,then X is a wγ-space.
Last, we consider the open closed images of γ-spaces or wγ-spaces.
Theorem 5.5. Let f : X −→ Y be an open closed map. Then the following
statements hold.
(1) If X is a γ, T1 -space, then Y is γ.
(2) If X is a wγ, T1 -space, then Y is wγ.
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[2] Ceder J. G, Some generalizations of metric spaces, Pacific J. Math., 11(1961),
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[17] Hodel R. E, Spaces defined by sequences of open covers which guarantee thatcertain sequences have cluster points, Duke Math. J., 39(1972), 253-263.
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Daigaku, Sec. A, 12(1973), 46-48.
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Education, Gunma Univ., 50(2002), 79-84.
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Department of Mathematics, Okayama University, Tsushima, Okayama, Japan
E-mail address: yoshiopka@math.okayama-u.ac.jp

参加者リスト（五十音順）
新井 達也
池田 義人
伊藤 宗彦
今井 淳
大田 春外
奥山 晃弘
小野 仁
加藤 久男
川口 慎二
木村 孝
家本 宣幸
小竹 義朗
菰田智恵子
小山 晃
酒井 克郎
酒井 政美
嶋根 紀仁
清水 一生
Dmitri Shakhmatov
鈴木 晃
高橋 真
田中 秀典
知念 直紹
千葉 慶子
寺田 敏司
Artur H. Tomita
友安 一夫
新田 眞一
平田 康史
深石 博夫
細渕 昌美
渕野 昌
Joerg Brendle
保科 隆雄
堀内 清光
牧 春夫
三村 護
宮嵜 和美
宮田 任寿
三輪 拓夫
牟 磊（Mou Lei）
森下 和彦
矢ヶ崎 達彦矢口 雅人
矢島 幸信
矢田部 俊介arai@math.tsukuba.ac.jp
yoshito@u-gakugei.ac.jp
ito@is.kochi-u.ac.jp
aimai@math.kyushu-u.ac.jp
echohta@ipc.shizuoka.ac.jp
okuyama@kobe-u.ac.jp
tsjono@eng.shizuoka.ac.jp
hisakato@sakura.cc.tsukuba.ac.jp
kimura@post.saitama-u.ac.jp
nkemoto@cc.oita-u.ac.jp
kotake@edu.gunma-u.ac.jp
komodac@fne.freeserve.ne.jp
koyama@cc.osaka-kyoiku.ac.jp
sakaiktr@sakura.cc.tsukuba.ac.jp
msakai@cc.kanagawa-u.ac.jp
shimane@kagoshima-ct.ac.jp
issei@kurt.scitec.kobe-u.ac.jp
dmitri@dpc.ehime-u.ac.jp
sakira@kobe-u.ac.jp
makoto@kobe-u.ac.jp
htanaka@cc.osaka-kyoiku.ac.jp
naochin@math.tsukuba.ac.jp
smktiba@ipc.shizuoka.ac.jp
terada@math.sci.ynu.ac.jp
tomita@ime.usp.brtomoyasu@cc.miyakonojo-nct.ac.jp
nitta@sg-jc.ac.jp
yhira@jb3.so.ne.jp
fukaishi@ed.kagawa-u.ac.jp
mhsbc@kasei-gakuin.ac.jp
fuchino@isc.chubu.ac.jp
brendle@kurt.scitec.kobe-u.ac.jp
takaohsn@math.tsukuba.ac.jp
horiuchi@konan-u.ac.jp
makih@pop12.odn.ne.jp
mimura@math.okayama-u.ac.jp
BZQ22206@nifty.ne.jp
tmiyata@kobe-u.ac.jp
miwa@math.shimane-u.ac.jp
mrst@ashitech.ac.jp
yagasaki@ipc.kit.ac.jp
yuki@cc.kanagawa-u.ac.jp
yatabe@kobe.email.ne.jp

筑波技術短期大学
東京学芸大学
高知大学
京都大学（研究生）
静岡大学
神戸女子短期大学
静岡大学
筑波大学
筑波大学（院生）
埼玉大学
大分大学
群馬大学
静岡大学（院生）
大阪教育大学
筑波大学
神奈川大学
鹿児島高専
神戸大学（院生）
愛媛大学
神戸大学
神戸大学
大阪教育大学
筑波大学
静岡大学
横浜国立大学
Sao Paulo University
都城高専
四条畷学園短期大学
筑波大学（院生）
香川大学
東京家政学院大学
中部大学
神戸大学
筑波大学
甲南大学
佐賀大学
岡山大学
大阪電通大学
神戸大学
島根大学
静岡大学（研究生）
足利工業大学京都工繊大学
筑波大学（院生）
神奈川大学
神戸大学（院生）

参加者リスト（五十音順）（つづき）
山内 貴光
山崎 薫里
山田 耕三
吉岡 巌
渡辺 正

kaori@math.tsukuba.ac.jp
eckyama@ipc.shizuoka.ac.jp
yoshioka@math.okayama-u.ac.jp
tadashi@yamaguchi-u.ac.jp

筑波大学（院生）
筑波大学
静岡大学
岡山大学
山口大学

Participants
Tatsuya Arai
Keiko Chiba
Naotsugu Chinen
Sakaé Fuchino
Hiroo Fukaishi
Yasushi Hirata
Kiyomitsu Horiuchi
Takao Hoshina
Masami Hosobuchi
Yoshito Ikeda
Atsushi Imai
Munehiko Ito
Brendle Joerg
Hisao Kato
Shinji Kawaguchi
Nobuyuki Kemoto
Takashi Kimura
Chieko Komoda
Yoshiro Kotake
Akira Koyama
Mou Lei
Haruo Maki
Mamoru Mimura
Takahisa Miyata
Kazumi Miyazaki
Takuo Miwa
Kazuhiko Morishita
Shinichi Nitta
Haruto Ohta
Teruhiko Okuyama
Jin OnoKatsuro Sakai
Masami Sakai
Norihito Shimane
Issei Shimizu
Dmitri Shakhmatov
Akira Suzuki
Makoto Takahashi
Hidenori Tanaka
Toshiji Terada
Artur H. Tomita
kazuo Tomoyasu
Tatsuhiko Yagasaki
Masato Yaguchi
Yukinobu Yajima
Shunsuke Yatabe

arai@math.tsukuba.ac.jp
smktiba@ipc.shizuoka.ac.jp
naochin@math.tsukuba.ac.jp
fuchino@isc.chubu.ac.jp
fukaishi@ed.kagawa-u.ac.jp
yhira@jb3.so.ne.jp
horiuchi@konan-u.ac.jp
takaohsn@math.tsukuba.ac.jp
mhsbc@kasei-gakuin.ac.jp
yoshito@u-gakugei.ac.jp
aimai@math.kyushu-u.ac.jp
ito@is.kochi-u.ac.jp
brendle@kurt.scitec.kobe-u.ac.jp
hisakato@sakura.cc.tsukuba.ac.jp

Tsukuba Junior College of Tech
Shizuoka Univ
Tsukuba Univ
Chunu Univ
Kagawa Univ
Tsukuba Univ (grad student)
Konan Univ
Tsukuba Univ
Tokyo Kasei Gakuin Univ
Tokyo Gakugei Univ
Kyoto Univ（Research Scholar）
Kochi Univ
Kobe Univ
Tsukuba Univ
Tsukuba Univ (grad student)
nkemoto@cc.oita-u.ac.jp
Oita Univ
kimura@post.saitama-u.ac.jp
Saitama Univ
komodac@fne.freeserve.ne.jp
Shizuoka Univ (grad student)
kotake@edu.gunma-u.ac.jp
Gumma Univ
koyama@cc.osaka-kyoiku.ac.jp
Osaka Kyoiku Univ
Shizuoka Univ (Research Scholar)
makih@pop12.odn.ne.jp
Saga Univ
mimura@math.okayama-u.ac.jp
Okayama Univtmiyata@kobe-u.ac.jp
Kobe Univ
BZQ22206@nifty.ne.jp
Osaka Dentsu Univ
miwa@math.shimane-u.ac.jp
Shimane Univ
mrst@ashitech.ac.jp
Ashikaga Inst of Technitta@sg-jc.ac.jp
Shijonawate-Gakuen Junior College
echohta@ipc.shizuoka.ac.jp
Shizuoka Univokuyama@kobe-u.ac.jp
Kobe Women Junior College
tsjono@eng.shizuoka.ac.jp
Shizuoka Univ
sakaiktr@sakura.cc.tsukuba.ac.jp
Tsukuba Univ
msakai@cc.kanagawa-u.ac.jp
Kanagawa Univ
shimane@kagoshima-ct.ac.jp
Kagoshima College of Tech
issei@kurt.scitec.kobe-u.ac.jp
Kobe Univ (grad student)
dmitri@dpc.ehime-u.ac.jp
Ehime Univ
sakira@kobe-u.ac.jp
Kobe Univ
makoto@kobe-u.ac.jp
Kobe Univ
htanaka@cc.osaka-kyoiku.ac.jp
Osaka Kyoiku Univ
terada@math.sci.ynu.ac.jp
Yokohama National Univ
tomita@ime.usp.br
Sao Paulo University
tomoyasu@cc.miyakonojo-nct.ac.jp Miyakonojo College of Techyagasaki@ipc.kit.ac.jp
Kyoto Inst of Tech
Tsukuba Univ (grad student)
yuki@cc.kanagawa-u.ac.jp
Kanagawa Univ
yatabe@kobe.email.ne.jp
Kobe Univ (grad student)Participants (continue)
Takamitsu Yamauchi
Kaori Yamazaki
kaori@math.tsukuba.ac.jp
Kohzo Yamada
eckyama@ipc.shizuoka.ac.jp
Iwao Yoshioka
yoshioka@math.okayama-u.ac.jp
Tadashi Watanabetadashi@yamaguchi-u.ac.jp

Tsukuba Univ (grad student)
Tsukuba Univ
Shizuoka Univ
Okayam Univ
Yamaguchi Univ