Mem. Fac. Sci. Eng. Shimane Univ.
Series B: Mathematical Science
33 (2000), pp.11–19

SOME QUESTIONS ON TRANSFINITE DIMENSIONS

Yasunao Hattori
Dedicated to Professor Maretsugu Yamasaki for his 60th birthday

Abstract. In the present note, we summarize open questions arising from recent
research in transﬁnite dimensions.

1. Introduction
In the present note, we shall consider some open questions on transﬁnite dimensions. Some of them are already asked by the author in the literatures.
We denote trInd (trind) by large (small) transﬁnite inductive dimension which
is a natural transﬁnite extension of large (small) inductive dimension Ind (ind).
A normal space X is called strongly countable-dimensional if X is the union ofcountably many closed subsets Xn , n = 1, 2, . . . , with dim Xn < ∞. For eachordinal number α, we write α = λ(α) + n(α), where λ(α) is a limit ordinal andn(α) is a ﬁnite ordinal. For a normal space X and a non-negative integer n, weput
Pn (X) =∪{U : U is an open set of X such that dim U ≤ n}.

Let X be a normal space and α be either an ordinal number or the integer −1.
The strong small transﬁnite dimension sind of X is deﬁned as follows ([B]):(i) sind X = −1 if and only if X = ∅.
∪(ii) sind X ≤ α if X is expressed in the form X = {Pξ : ξ < α}, where∪Pξ = Pn(ξ) (X \ {Pη : η < λ(ξ)}).
If sind X ≤ α for some α, we say that X has strong small transﬁnite dimension.
1991 Mathematics Subject Classiﬁcation. Primary 54F45; secondary 54C10, 54E35.
Key words and phrases. Large transﬁnite dimension, small transﬁnite dimension, strong
small transﬁnite dimension, order dimension, strongly countable-dimensional space, ﬁnitistic
space, K-approximation, metrizable space.
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Recall from [H4] that a normal space X has strong large transﬁnite dimension
if X has both large transﬁnite dimension and strong small transﬁnite dimension. The following characterization of spaces that have strong large transﬁnite
dimension is useful.
Proposition A [H3, Propositions 2.2 and 2.3]. Let X be a metrizable space.
Then X has strong large transﬁnite dimension if and only if X is ﬁnitistic andstrongly countable-dimensional.
The concept of ﬁnitistic spaces was introduced by Swan [Sw] for working inﬁxed point theory and is applied to the theory of transformation groups by using
the cohomological structures (cf. [AP]). For a family U of a space X the order
ord U of U is deﬁned as follows: ordx U = |{U ∈ U : x ∈ U }| for each x ∈ X andord U = sup{ordx U : x ∈ X}. We say a family U has ﬁnite order if ord U = n forsome natural number n. A space X is said to be ﬁnitistic if every open cover of
X has an open reﬁnement with ﬁnite order. We notice that ﬁnitistic spaces arealso called boundedly metacompact spaces (cf. [FMS]). It is clear that all compact
spaces and all ﬁnite dimensional paracompact spaces are ﬁnitistic spaces. Moreprecisely, we have a useful characterization of ﬁnitistic spaces.
Proposition B [H2], [DMS]. A paracompact space X is ﬁnitistic if and only
if there is a compact subspace C of X such that dim F < ∞ for every closed
subspace F with F ∩ C = ∅.
The dimension-theoretic properties of ﬁnitistic spaces are investigated by several authors (cf. [DP], [DS], [DT], [DMS] [H2] and [H6]).
We denote the set of natural numbers by N. We refer the reader to [E] and[N] for basic results in dimension theory.
2. Questions arising from characterizations
by means of K-approximations
In [DMS], Dydak-Mishra-Shukla introduced a concept of a K-approximationof a mapping to a metric simplicial complex and characterized n-dimensional
spaces and ﬁnitistic spaces in terms of K-approximations. Let X be a space, Ka metric simplicial complex and f : X → K a continuous mapping. A mappingg : X → K is said to be a K-approximation of f if for each simplex σ ∈ Kand each x ∈ X, f (x) ∈ σ implies g(x) ∈ σ. A K-approximation g : X → K
of f is called an n-dimensional K-approximation if g(X) ⊂ K (n) and a ﬁnite
dimensional K-approximation if g(X) ⊂ K (m) for some natural number m, whereK (m) denotes the m-skeleton of K. We have the following characterizations ofinﬁnite-dimensional spaces by means of K-approximations in [H5]. For a space

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X we denote
D(X) = {D : D is a closed discrete subset of X}.
Theorem 2.1 [H5, Theorem]. For a metrizable space X the following are equivalent.
(a) X has strong large transﬁnite dimension.
(b) There is a function ϕ : D(X) → ω such that for every metric simplicial
complex K and every continuous mapping f : X → K there is a Kapproximation g of f such that g(D) ⊂ K (ϕ(D)) for each D ∈ D(X).
(c) For every integer m ≥ −1 there is a function ψ : D(X) → ω such thatfor every metric simplicial complex K and every continuous mapping f :X → K there is a ﬁnite dimensional K-approximation g of f such thatg(D) ⊂ K (ψ(D)) for each D ∈ D(X) and g|f −1 (K (m) ) = f |f −1 (K (m) ).
Theorem 2.2 [H5, Corollary]. For a paracompact space X the following are
equivalent.
(a) X is a strongly countable-dimensional space.
(b) There is a function ϕ : X → ω such that for every metric simplicial
complex K and every continuous mapping f : X → K there is a Kapproximation g of f such that g(x) ∈ K (ϕ(x)) for each x ∈ X.
(c) For every integer m ≥ −1 there is a function ψ : X → ω such thatfor every metric simplicial complex K and every continuous mapping f :X → K there is a K-approximation g of f such that g(x) ∈ K (ψ(x)) for
each x ∈ X and g|f −1 (K (m) ) = f |f −1 (K (m) ).
Concerning the Theorems 2.1 and 2.2, we can ask the following.
Question 2.1. Are the conditions (a) and (b) in Theorem 2.1 equivalent forparacompact spaces?
We have a simple answer the question, i.e., the implication (b) ⇒ (a) does nothold. In fact, for each m, n ∈ N with m ≤ n, Vopěnka [Vo] constructed a compact
space Xm,n such that dim Xm,n = m and Ind Xm,n = n. Let X be the topological⊕∞sum n=1 X1,n of X1,n , n ∈ N. Then X does not have large transﬁnite dimension(and hence X does not satisfy (a)). Since dim X = 1, it follows from [DMS] thatfor every metric simplicial complex K and every continuous mapping f : X → K
there is a 1-dimensional K-approximation g of f . Hence X satisﬁes the condition(b).
Now, we consider the following condition which is weaker than (a).

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YASUNAO HATTORI(a’) X is a strongly countable-dimensional space satisfying the following condition (K) (cf. [P]):(K) There is a compact subspace C of X such that dim F < ∞ for every
closed subspace F of X with F ∩ C = ∅.
We consider the relations between (a), (b) in Theorem 2.1 and (a’) for normal(paracompact) spaces.
In [E, §7.3], Engelking reformulated the class of spaces that have strong smalltransﬁnite dimension by use of a new dimension function transﬁnite dimensional
kernel trker. He called a space that has transﬁnite dimensional kernel as a shallow
space. One should notice that a normal space X is a shallow space if and only
if X has strong small transﬁnite dimension and sind X = trker X if sind X is alimit ordinal and sind X = trker X + 1 otherwise.
We shall consider four implications separately.
I. (a) ⇒ (a’).
Fact 2.1 ([E, Theorem 7.1.23]). If a weakly paracompact, strongly hereditarily normal space X has large transﬁnite dimension Ind X, then X satisﬁes thecondition (K).
Fact 2.2 ([E, Theorem 7.3.13]; [H1, Theorem 1.2] for metrizable spaces). If
a weakly paracompact perfectly normal shallow space X, then X is a strongly
countable-dimensional space.
We can ask the following.
Question 2.2. Can we drop the perfectness in Fact 2.2? I.e., is a weakly paracompact normal shallow space a strongly countable-dimensional?
We have a partial answer the question.
Theorem 2.3. Let X be a hereditarily weakly paracompact and hereditarily normal space. If X is a shallow space, then X is a strongly countable-dimensional
space.
Proof. We show by the transﬁnite induction on sind X = α.
Case 1. Suppose that α is a limit ordinal number. We notice that X is expressed∪∪in the form X = {Pξ : ξ < α}, where Pξ = Pn(ξ) (X \ {Pη : η < λ(ξ)}). We∪
put Gξ = {Pη : η < ξ} for ξ < α. Then {Gξ : ξ < α} is an open covering of
X and sind Gξ ≤ ξ < α. By the inductive assumption, Gξ is strongly countabledimensional for each ξ < α. Hence it follows from [E, Theorem 5.2.17] that X isstrongly countable-dimensional.

SOME QUESTIONS ON TRANSFINITE DIMENSIONS

15∪
Case 2. Suppose that α = β + 1. Let Y = X \ {Pξ : ξ < λ(α)}).
Then sind Y ≤ β < α. By the inductive assumption, Y is strongly countabledimensional. Hence there is a countable cover {F1 , F2 , . . . } of Y by ﬁnite dimensional closed sets. Since Pα is a closed set of X such that X = Y ∪ Pα anddim Pα = n(α) < ∞. We put Ei = Fi ∪ Pα for each i ∈ N. Then it follows
that Ei closed in X and dim Ei ≤ max{dim Fi , dim Pα }. Hence X is strongly
countable-dimensional. ¤Corollary 2.1. Let X be a hereditarily weakly paracompact, strongly hereditarily
normal space. Then the implication (a) ⇒ (a’) holds.
Question 2.2’. Do Theorem 2.3 and the Corollary 2.1 hold for weakly paracompact normal spaces?
II. (a’) ⇒ (a).
It is known that a normal space X is a shallow space if and only if every
non-empty closed subspace F of X contains a non-empty normal open subspace
U of F such that dim U < ∞ ([E, Problem 7.3.A]). Hence, by the Baire category
theorem, every normal Čech-complete, strongly countable-dimensional space is ashallow space. This implies the following.
Proposition 2.1. Let X be a normal space satisfying the condition (K). If X isa strongly countable-dimensional space, then X is a shallow space.
Proof. Let C be a compact subspace of X such that dim F < ∞ for every closed
subspace F of X with F ∩ C = ∅. Then C is a compact strongly countabledimensional space. Hence C is shallow and hence X is a shallow space by [E,Problem 7.3.H]. ¤As we mentioned above, there is a paracompact space X such that dim X = 1,
but X does not have large transﬁnite dimension. This example leads the Indversion of the condition (a’). A normal space X is strongly countable-dimensionalwith respect to Ind (shortly s.c.d.-Ind) if X is a union of countably many closed
subspaces Xn , n ∈ N, such that Ind Xn < ∞ for each n ∈ N. Further, we
introduce a notion similar to the condition (K).
(K-Ind) There is a compact subspace C of X such that Ind F < ∞ for every closed
subspace F of X with F ∩ C = ∅.
We consider the following.
(a”) X is an s.c.d.-Ind space satisfying the condition (K-Ind).
Then we have

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Proposition 2.2. Let X be a hereditarily normal space. If X is an s.c.d.-Ind
space satisfying the condition (K-Ind), then X has large transﬁnite dimension
Ind X.
Proof. Let C be a compact subspace of X such that Ind F < ∞ for every closed
subspace F of X with F ∩ C = ∅. Since C is an s.c.d.-Ind compact space, by[F, Theroems 1, 3], C has large transﬁnite dimension. Then it follows from [E,Lemma 7.1.24] that X has large transﬁnite dimension and Ind X ≤ ω0 +Ind C. ¤Question 2.3. Does Proposition 2.2 hold for normal spaces?
Question 2.4 [F, Problem 3]. Does every compact space which can be represented as the union of countably many subspaces which all have large transﬁnite
dimension have itself large transﬁnite dimension?
We notice that if Question 2.4 has an aﬃrmative answer, then Question 2.3
does.
III. (a’) ⇒ (b).
By the proof of Theorem 2.1 (see [H5]), we have the following.
Proposition 2.3. Let X be a strongly countable-dimensional paracompact space.
If there is a compact subspace C of X such that C has a countable character and
Ind F < ∞ for every closed subspace F of X with F ∩ C = ∅, then X satisﬁes
the condition (b).
We do not know the proposition above holds for every strongly countabledimensional paracompact space satisfying the condition (K).
IV. (b) ⇒ (a’).
The proof of (b) ⇒ (a) of Theorem 2.1 works well for paracompact spaces and it
shows that the condition (b) implies the condition (a’) ([H5]). (The metrizability
is used for the equivalence between (a) and (a’) in Theorem 2.1). Hence theimplication (b) ⇒ (a’) holds for every paracompact space X.
We turn our attention to ﬁnitistic spaces. In [H6], we proved that there is auniversal space L(τ ) for the class of ﬁnitistic metrizable spaces with the weight≤ τ and asked if {h : h : X → L(τ ) is a homeomorphic embedding} is dense in
the function space C(X, L(τ )) ([H6, Question 1]). Quite recently, Kulesza answers
the question negatively. The following question asked in the same paper still
remains open.

SOME QUESTIONS ON TRANSFINITE DIMENSIONS

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Question 2.5 [H6, Question 2]. Let X be a metrizable (or paracompact) spacesuch that {f ∈ C(X, I ω ) : f (X) \ f (X) is countable-dimensional} is residual inC(X, I ω ). Is X ﬁnitistic?
3. Questions on order dimension
In this section, we shall consider a transﬁnite dimension deﬁned by an orderof closed mappings. In 1955, K. Morita [M] proved a fundamental theorem on
the dimension and closed mappings in metrizable spaces : For a metrizable space
X dim X ≤ n if and only if there are a metrizable space Z with dim Z = 0 and aclosed mapping f of Z onto X such that every ﬁber of f contains at most n + 1
points. The theorem has many applications to inﬁnite dimensional spaces. We
begin with some deﬁnitions.
Deﬁnition 3.1 [B2]. Let L be a set, Fin L the collection of all non-empty ﬁnite
subsets of L and M a subset of Fin L. Let α > 0 be an ordinal number. For
σ ∈ {∅} ∪ Fin L, we put
M σ = {τ ∈ Fin L : σ ∪ τ ∈ M and σ ∩ τ = ∅}.
If σ = {a}, we write M σ = M a . Then the order Ord M of M is inductively
deﬁned as follows:(1) Ord M = 0 if M = ∅.
(2) Ord M ≤ α if Ord M a < α for each a ∈ L.
If Ord M ≤ α for some ordinal number α, we say that OrdM exists (or M has
Ord M ).
We notice that Ord M ≤ n iﬀ |σ| ≤ n for each σ ∈ M , where n < ω.
By use of this order, F. G. Arenas ([A]) deﬁned an order of mappings as follows.
Deﬁnition 3.2 [A]. Let X and Y be topological spaces and f : X → Y amapping. Let T (X) be the topology of X. We put
O(f ) = {τ = {U1 , . . . , Un } ∈ FinT (X) :Ui ∩ Uj = ∅ for i 6= j andn∩f (Ui ) 6= ∅}.

i=1Then the order Ord f of f is deﬁned as Ord f = Ord O(f ).
By use of the transﬁnite order of mappings, Arenas extended the covering
dimension to a transﬁnite dimension as follows.

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Deﬁnition 3.3 [A]. Let X be a Tychonoﬀ space and α an ordinal number. Then
X has the order dimension O-dim X ≤ α if and only if there are a strongly zerodimensional space Z and a perfect mapping f of Z onto X such that Ord f ≤α + 1. We say that X has the order dimension O-dim X (or O-dim X exists) if
O-dim X ≤ α for some ordinal number α.
The following is a characterization of spaces that have large transﬁnite dimension in terms of ﬁnite-to-one closed continuous mappings.
Theorem 3.1 [HY, Theorem 2.7]. A metrizable space X has the order dimension
O-dim X if and only if X has large transﬁnite dimension Ind X.
Connects with the theorem, we can ask the following.
Question 3.1. Does the theorem hold for every paracompact space?
A normal space X is called countable-dimensional if X is the union of countably
many subspaces Xn , n = 1, 2, . . . , with dim Xn < ∞.
Question 3.2 [HY, Question 2.9]. Let X be a compact space having the orderdimension O-dim X. Is X countable-dimensional?
Remark. In [A], Arenas mentioned that there is a compact space X having theorder dimension O-dim X, but X is not countable-dimensional. However, his
proof is not correct (see [HY, Remark 2.8]). Hence the question above still seems
to be open.
We further the investigation of the relation between the order dimension andlarge transﬁnite dimension.
Theorem 3.2 [HY, Theroem 3.5]. Let X be a metrizable space having the orderdimension O-dim X. Then the inequality Ind X ≤ O-dim X holds.
Concerning the result above, we can ask the following:
Question 3.3 [HY, Question 3.6]. Let X be a metrizable space having largetransﬁnite dimension Ind X. Does the inequality O-dim X ≤ Ind X hold?
We notice that Ind S α = O-dim S α = α holds for every ordinal number α < ω1 ,where S α denotes the Smirnov’s compactum ([HY, Theorem 3.8]).
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Department of Mathematics, Shimane University, Matsue, 690-8504 JAPANe-mail: hattori@math.shimane-u.ac.jp