Math. J. Okayama Univ. 50 (2008), 177–199

K-SEMIMETRIZABILITIES AND C-STRATIFIABILITIESOF SPACES
Iwao YOSHIOKA

1. Introduction and definitions
In 1966, Arhangel’skiı̆ [1] introduced the concepts of symmetrizable spaces
and he showed that a T2 -space is metrizable if, and only if, it has a compatible symmetric d satisfying condition (A): d(F, K)>0 for any disjoint closed
subset F and compact subset K. Also, Arhangel’sk ĭi gave the class of spaces
with a compatible symmetric d satisfying condition (K): d(H, K)>0 for any
disjoint compact subsets H and K, and he conjectured that every symmetrizable space has a compatible symmetric satisfying condition (K). After
that, in 1975, Martin [26] presented the question on whether every regularsemimetrizable space is K-semimetrizable (i.e. it has a compatible semimetric satisfying condition (K)) , or if every Moore space is K-semimetrizable.
In 1979, Burke [6] gave a negative answer that there exists a separable Moore
space which is not K-semimetrizable.
Lee [22] defined the class of c-stratifiable spaces which contains the classesof spaces with a regular Gδ -diagonal and of γ, T2 -spaces. He proved that aspace X is K-semimetrizable if, and only if, X is c-stratifiable semimetrizable if, and only if, X is regular c-stratifiable, first countable and β. On theother hand, in [31], we introduced the concepts of strong α-ness and showedthat every strongly α, wM -space is metrizable. The properties of strongly
α-spaces were also studied in the same paper.
In this note, we study the relations among c-stratifiable spaces, strongly αspaces, K-semimetrizable spaces, developable spaces and Nagata spaces, andthe conditions for spaces to be K-semimetrizable or full K-semimetrizable.
We prove that a space X is K-semimetrizable if, and only if, it is a cstratifiable q, β-space. We also show that a space X is full K-semimetrizableif, and only if, it is a wθ, β-space with a regular G δ -diagonal, which is aslight generalization of [32; Theorem 2]. We also show that a space X is
Nagata if, and only if, it is K-semimetrizable wcc if, and only if, it is regularsemimetrizable wcc. Moreover, for metrizations of wM -spaces, we have thatevery wM -space with a G∗δ -diagonal is metrizable.
In §2, we study the relations between c-stratifiable spaces and strongly αspaces. Also, we consider the conditions for spaces to be strongly α orc-stratifiable. In particular, we show that in the realm of c-stratifiable
spaces, wN -spaces are Nagata, q-spaces are first countable, wcc-spaces are
k-semistratifiable and w∆-spaces are developable.
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I. YOSHIOKAIn §3, we study the class of K-semimetrizable spaces. First, we showthat a space X is K-semimetrizable if, and only if, it is c-stratifiable q,β. Secondly, we prove that in the class of pseudocompact spaces or locally connected rim-compact spaces, developable K-semimetrizable spaces
are equivalent to c-stratifiable β-spaces (or K-semimetrizable spaces), andevery metacompact p-space with a Gδ -diagonal is a K-semimetrizable Moore
space.
In §4, for the class of wθ, wcc-spaces which contains the class of wM spaces, we show that every wθ, wcc-space with a G ∗δ -diagonal is metrizableand every c-stratifiable wθ, wcc-space is metrizable.
Throughout this paper, we assume that all spaces are T 1 , but paracompactness is assumed to be T2 . We denote a sequence {xn |n ∈ N} by {xn }and the set of natural numbers by N. Finally, we refer the reader to [9] for
undefined terms.
Definition 1.1. A g-f unction on a space X with a topology T is a map
g : N×X −→ T such that g(n, x) = gn (x) is an open neighbourhood of x forevery x ∈ X and each n ∈ N and we denote the map g by ({g n (x)}|x ∈ X).
For a subset A of X, we put gn (A) = ∪{gn (x)|x ∈ A}.
A point p in X is called a cluster point of a sequence {x n } ⊂ X if any
open neighbourhood of p contains xn for infinitely many n’s.
For a space X, we now consider the following conditions on a g-function({gn (x)}|x ∈ X).
(A) If gn (x) ∩ gn (xn ) 6= ∅ (n ≥ 1), then x is a cluster point of {x n }.
(B) If gn (x) ∩ gn (xn ) 6= ∅ (n ≥ 1), then {xn } has a cluster point.
(C) If x ∈ gn (xn ) (n ≥ 1), then x is a cluster point of {x n }.
(D) If x ∈ gn (xn ) (n ≥ 1), then {xn } has a cluster point.
(E) If yn ∈ gn (xn ) (n ≥ 1) and {yn } has a cluster point, then {xn } has acluster point.
(F) If xn ∈ gn (x) (n ≥ 1), then {xn } has a cluster point.
(G) If yn ∈ gn (p), xn ∈ gn (yn ) (n ≥ 1), then p is a cluster point of {x n }.
(H) If yn ∈ gn (p), xn ∈ gn (yn ) (n ≥ 1), then {xn } has a cluster point.
(I) If yn ∈ gn (p), xn , p ∈ gn (yn ) (n ≥ 1), then p is a cluster point of {x n }.
(J) If yn ∈ gn (p), xn , p ∈ gn (yn ) (n ≥ 1), then {xn } has a cluster point.
(K) If xn , p ∈ gn (yn ) (n ≥ 1), then p is a cluster point of {x n }.
(L) If xn , p ∈ gn (yn ) (n ≥ 1), then {xn } has a cluster point.
In the above conditions (A)-(L), we can assume that g n+1 (x) ⊂ gn (x) for
every x ∈ X and each n ∈ N.

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Definition 1.2. A space with a g-function satisfying (A) is called a N agata
space [15] (Nagata spaces were first defined by Ceder [7]) and a space witha g-function satisfying (B) is called a wN -space [18]. In this case the gfunction is called a N agata-f unction (a wN -f unction, respectively).
Definition 1.3. A space X is called a semistratif iable (β-, wcc (=weak
contraconvergent)-, q-, γ-, wγ-, θ-, wθ-) space if X has a g-function satisfying (C) ( (D), (E), (F), (G), (H), (I), (J), respectively). (See [17], [18] and[31])
The following result is not difficult to see.
Proposition 1.4. [31; Theorem 3.5] A space X is wN if, and only if, it isq and wcc.
Definition 1.5. A space X is called stratif iable [3] (equivalently, M 3 [7])if X has a g-function that satisfies (C) and if x ∈/ g m (F ) for some m ∈ N,
whenever F is closed and x ∈/ F . The class of k-semistratif iable spaces
introduced by Lutzer [24] can be characterized by the following conditions[12, 31]. A space X is k-semistratifiable if, and only if, X has a g-function({gn (x)}|x ∈ X) such that gm (F ) ∩ K = ∅ for some m ∈ N, whenever F isclosed, K is compact and F ∩K = ∅, if, and only if, in the class of T 2 -spaces,
X has a g-function ({gn (x)}|x ∈ X) such that whenever yn ∈ gn (xn ) (n ≥ 1)
and {yn } −→ y, then {xn } −→ y.
The following implications are known.
Nagata =⇒ stratifiable =⇒ k-semistratifiable =⇒ semistratifiable =⇒ β.
Also, it is known that a Nagata space is equivalent to a first countablestratifiable space and every stratifiable space is paracompact. Every semistratifiable space X is subparacompact and has a G δ -diagonal if it is T2 [14;Theorem 5.11].
2. c-stratifiable spaces and strongly α-spaces
We begin by considering the relations between c-stratifiable spaces andstrongly α-spaces, and the conditions for spaces to be c-stratifiable orstrongly α.
Definition 2.1. A space X is called c-stratifiable [22] (c-semistratifiable[25]) if X has a g-function such that if x ∈/ K, where K is compact, then
/ gm (K); in [25], it is assumed that K is closed compact) forx∈/ gm (K) (x ∈some m ∈ N. A space X is called cs-stratif iable if X has a g-function such

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that if x ∈/ C, where C is the union of a convergent sequence and any oneof its limit points, then x ∈/ gm (C) for some m ∈ N. A space X is called
weak c-stratif iable if X has a g-function such that, whenever C and D are
disjoint compact subsets, then gm (C) ∩ D = ∅ for some m ∈ N.
Every stratifiable space or γ, T2 -space is c-stratifiable [22], but the Sorgenfrey line is a paracompact c-stratifiable space which is not semistratifiable.
Also, every c-stratifiable space is cs-stratifiable and weak c-stratifiable [22;Theorem 1.3], every cs-stratifiable space is T 2 and every semistratifiable T2 space is c-semistratifiable.
Definition 2.2. A space X is called strongly α [31] (α [17]) if X has ag-function such that (i) for each n ∈ N, y ∈ g n (x) =⇒ gn (y) ⊂ gn (x) and(ii) ∩n≥1 gn (x) = {x} (∩n≥1 gn (x) = {x}).
For g-functions in Definitions 2.1 and 2.2, we can assume that g n+1 (x) ⊂gn (x) for every x ∈ X and each n ∈ N. Every strongly α-space is also T 2 .
Theorem 2.3.
Every cs-stratifiable q-space X or regular weakc-stratifiable q-space X is a first countable c-stratifiable space.
Proof. Let g be a q and cs-stratifiable function of a space X. We showthat {gn (x)} is an open neighbourhood base of x for every x ∈ X. Supposethat x ∈ X and xn ∈ gn (x) \ U (n ≥ 1) for some open neighbourhood U of x.
Since g is a q-function, {xn } has a cluster point p and p ∈/ {x}. Since g is a
cs-stratifiable function, p ∈/ gm (x) ⊃ {xj |j ≥ m} 3 p for some m ∈ N. This
contradiction implies that {gn (x)} is a neighbourhood base of x. To see thatg is a c-stratifiable function, suppose that x ∈/ K, where K is compact in X,
and x ∈ ∩n≥1 gn (K). Then, there exist sequences {x n } ⊂ K and {yn } suchthat yn ∈ gn (x) ∩ gn (xn ). Since K is sequentially compact, {x n(i) } −→ p forsome point p ∈ K and some subsequence {x n(i) } ⊂ {xn }, and {yn(i) } −→ x
for the subsequence {yn(i) } ⊂ {yn }. Then x ∈/ {xn(i) |i ≥ 1} ∪ {p} = C,and hence x ∈/ gm (C) for some m ∈ N. Therefore, for some n(j) ≥ m,yn(j) ∈
/ gm (C) ⊃ gn(j) (xn(j) ) 3 yn(j) . This contradiction implies that g isalso a c-stratifiable function.
For the second part, we can assume that g is a weak c-stratifiable qfunction satisfying gn+1 (x) ⊂ gn (x). If x and y are distinct points, thengm (x) ∩ {y} = ∅ for some m ∈ N. Hence, ∩n≥1 gn (x) = {x}. Suppose thatx ∈ X and xn ∈ gn (x) \ U (n ≥ 1) for some open neighbourhood U of x.
Then {xn } has a cluster point p. Hence, p ∈ gn (x) for each n ∈ N. This
contradiction asserts that {gn (x)} is a neighbourhood base of x. Therefore,

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g is a c-stratifiable function by [22; Theorem 1.3].
Theorem 2.4. (1) Every strongly α-space X or k-semistratifiable T 2 -space
X is weak c-stratifiable.
(2) Every strongly α, q-space X is c-stratifiable.
Proof. (1): First, let g be a strongly α-function of X. Suppose that there
are disjoint compact subsets C and D such that x n ∈ gn (C) ∩ D (n ≥ 1).
Then xn ∈ gn (yn ) for some sequence {yn } ⊂ C. Hence {yn } clusters at apoint y ∈ C and contains a subsequence {y n(i) } such that yn(i) ∈ gi (y). Then
xn(i) ∈ gn(i) (yn(i) ) ⊂ gi (yn(i) ) ⊂ gi (y)(i ≥ 1), and {xn(i) } has a cluster pointx ∈ D. Then for each i ∈ N, x ∈ {xn(j) |j ≥ i} ⊂ gi (y). Therfore x = y,which is a contradiction. Next, that a k-semistratifiable T 2 -space is weakc-stratifiable follows from the equivalent condition of a k-semistratifiable
space in Definition 1.5.
(2): Let g be a q-function and h be a strongly α-function of a space X.
Here, we can assume that gn (x) ⊂ hn (x). For some x ∈ X and some compact subset K, suppose that x ∈/ K and x ∈ ∩ n≥1 gn (K). Then there exist
sequences {yn } and {zn } such that yn ∈ K and zn ∈ gn (x) ∩ gn (yn ). Lety ∈ K be a cluster point of {yn }. Then yn(i) ∈ gi (y) for some increasing subsequence {n(i)} of N. Also, since zn(i) ∈ gi (x)(i ≥ 1), {zn(i) } has a cluster
point z. Since yn(i) ∈ hi (y)(i ≥ 1), zn(i) ∈ gn(i) (yn(i) ) ⊂ hi (yn(i) ) ⊂ hi (y).
Therefore, {zn(j) |j ≥ i} ⊂ hi (y)(i ≥ 1) and hence, z ∈ hi (y)(i ≥ 1), which
implies that y = z. Moreover, since z n(i) ∈ gi (x) ⊂ hi (x)(i ≥ 1), we have{zn(j) |j ≥ i} ⊂ hi (x), and hence z ∈ hi (x). Consequently, x = z. This
contradiction implies that g is a c-stratifiable function.
We now study the conditions for spaces to be c-stratifiable or strongly α.
Definition 2.5. A space X is called a w∆-space [4] if it has a sequence {G n }of open covers such that whenever xn ∈ st(x, Gn ) (n ≥ 1), then {xn } has acluster point. A space X is called a developable space if it has a sequence{Gn } of open covers such that for each x ∈ X, the sequence {st(x, G n )} isa neighbourhood base of x. A regular developable space is called a M oore
space. These spaces are characterized by g-functions as follows [18]: A space
X is w∆ (developable) if and only if X has a g-function satisfying (L) ((K),
respectively).

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Definition 2.6. (1) For each k ∈ N, a space X is said to have a G δ (k)diagonal if X has a sequence {Gn } of open covers such that for any distinct points x and y, there exists m ∈ N such that y ∈/ st k (x, Gm ), where
stk+1 (x, Gm ) = st(stk (x, Gm )).
(2) A sequence {Gn } of open covers of a space X is said to satisfy the
3-link property [32] (equivalently, it is a G δ (3)-diagonal sequence) if it istrue that for any distinct points x and y, there exists m ∈ N such that nomember of Gm intersects both st(x, Gm ) and st(y, Gm ).
(3) A space X is said to have a regular G δ -diagonal [32] if there is asequence {Gn } of open covers of X such that if x and y are distinct pointsof X, then there are an integer m and open neighbourhoods U and V of x
and y, respectively, such that no member of G m intersects both U and V .
(4) A space X is said to have a G∗δ -diagonal if X has a sequence {Gn }of open covers such that whenever x 6= y, there exists m ∈ N that satisfiesy∈/ st(x, Gm ).
It is easily seen that for a sequence G = {G n } of open covers of a space
X, G is Gδ (2)-diagonal if, and only if, whenever x 6= y, there exists m ∈ N
satisfying x ∈/ st(p, Gm ) or y ∈/ st(p, Gm ) for every p ∈ X (this property iscalled strong Gδ -diagonal in [31]).
We note that for properties of a sequence {G n } of open covers of a space
X, the following implications hold:3-link property ⇒ regular Gδ -diagonal ⇒ G∗δ -diagonal ⇒ Gδ -diagonal and3-link property ⇒ Gδ (2)-diagonal = strong Gδ -diagonal ⇒ G∗δ -diagonal.
In the realm of paracompact spaces, these properties are all equivalent.
Every Nagata space is paracompact and has a G δ -diagonal. Every developable T2 -space has a Gδ (2)-diagonal and every regular semistraifiable space
has a G∗δ -diagonal [14, 17]. On the other hand, the space Ψ in Example 4.5
is a Moore space which does not have a regular G δ -diagonal.
Definition 2.7. (1) A space X is called orthocompact if every open coverof X has an open refinement V such that ∩W = ∩{W |W ∈ W} is open forevery W ⊂ V.
(2) A space X is called submetrizable if there is a continuous one-to-one
map from X onto a metric space.
It is well known that the following implications hold:metacompact spaces =⇒ orthocompact spaces, andstratifiable spaces =⇒ paracompact spaces with a G δ -diagonal [3, 29] =⇒submetrizable spaces .
Theorem 2.8. (1) Every space X with a regular G δ -diagonal is c-stratifiable.

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183(2) Every orthocompact space X with a G ∗δ -diagonal, or orthocompact regular space X with a Gδ -diagonal is strongly α.
(3) Every orthocompact developable T 2 -space X is strongly α and c-stratifiable.
(4) Every orthocompact regular c-semistratifiable β-space X is strongly α.
(5) Every submetrizable space X is strongly α and c-stratifiable.
Proof. (1) is proved in [22; Proposition 3.2].
(2): Let X be a regular space and let {G n } be a Gδ -diagonal sequenceof X. Then for each n ∈ N, Gn has an open refinement Hn such that
{H|H ∈ Hn } is a refinement of Gn and ∩W is open for every W ⊂ Hn .
Therefore, in both cases, we may assume that there exists a sequence {H n }of open covers such that(i) for each n ∈ N, ∩W is open for every W ⊂ H n , and(ii) for distinct points x and y, there is an m ∈ N such that x ∈ H ⊂ H
and y ∈
/ H for some H ∈ Hm .
Here, for any x ∈ X and each n ∈ N, we put h n (x) = ∩{H ∈ Hn |x ∈ H}.
Then the g-function ({hn (x)}|x ∈ X) satisfies the conditions of Definition
2.2.
(3): Since every developable T2 -space has a G∗δ -diagonal, X is a strongly
α from (2) and it is c-stratifiable from Theorem 2.4.
(4): Since every regular c-semistratifiable β-space is semistratifiable [25;
Theorem 3], it has a Gδ -diagonal. Hence X is strongly α from (2).
(5): Let f : X −→ M be a continuous one-to-one onto map, where M
is a metric space. By (3), M is strongly α and c-stratifiable. Therefore (5)follows from the following fact:
Let f : X −→ Y be a continuous one-to-one onto map. If h is a strongly
α-function (c-stratif iable f unction) of Y, then ({g n (x)}|x ∈ X), wheregn (x) = f −1 [hn (f (x))], is a strongly α-function (c-stratif iable f unction,
respectively) of X.
We note that every metacompact regular semistratifiable q-space is strongly α and hence it is c-stratifiable. Also, every regular k-semistratifiable qspace is Nagata [31], and hence it is strongly α, c-stratifiable. On the other
hand, the separable Moore space X in Example 4.6 is neither strongly α nor
c-stratifiable.
The following question arises naturally from (4) of the above Theorem.
Question 2.9. Is every paracompact (or metacompact regular) c-semistratifiable q-space, c-stratifiable ?

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Theorem 2.10. For a cs-stratifiable space X, the following implicationshold:(1) β =⇒ semistratif iable, (2) wcc =⇒ k-semistratif iable, (3) wN =⇒N agata, (4) wθ =⇒ θ, (5) wγ =⇒ γ and (6) w∆ =⇒ developable.
Proof. (1): Let g be a cs-stratifiable and β-function of X and let x ∈gn (xn ) (n ≥ 1). For any subsequence {xn(i) } of {xn }, {xn(i) } has a cluster point since x ∈ gi (xn(i) )(i ≥ 1). Let p be a cluster point of {x n }and p 6= x. Then {xn } contains a subsequence S = {xn(j) } such thatxn(j) ∈ gj (p) \ {x}(j ≥ 1). Since {xn(k) |k ≥ j} ⊂ gj (p)(j ≥ 1) and{p} = ∩j≥1 gj (p), p is the only cluster point of S. Hence S converges top. Since x ∈/ C = S ∪ {p}, x ∈/ gm (C) for some m ∈ N. But, for somen(k) ≥ m, x ∈ gn(k) (xn(k) ) ⊂ gm (C). This contradiction implies that x = p
is a cluster point of {xn }.
(2): Let g be a cs-stratifiable, wcc-function satisfying {x} = ∩ n≥1 gn (x)for every x ∈ X. Since X is a T2 -space, it is enough to show that g satisfies
the k-semistratifiable condition of Definition 1.5. Let y n ∈ gn (xn ) (n ≥1) and {yn } −→ y. First, we show that {xn } contains a subsequencewhich converges to y. Indeed, for any subsequence {x n(i) } of {xn }, yn(i) ∈gn(i) (xn(i) ) ⊂ gi (xn(i) )(i ≥ 1) and {yn(i) } −→ y, hence {xn(i) } has a cluster point. Let p be a cluster point of {x n }. It is easily seen that there
exists a subsequence S = {xn(i) } of {xn } such that xn(i) ∈ gi (p). Since{xn(j) |j ≥ i} ⊂ gi (p) for each i ∈ N, p is a unique cluster point of S. Hence
S converges to p. If p 6= y, then y ∈/ {xn(i) |i ≥ m}∪{p} = C for some m ∈ N.
Therefore y ∈/ gn(k) (C) for some k ≥ m. This contradiction asserts that S
converges to y. Next, if {xn } does not converge to y, then we have an open
neighbourhood W of y and a subsequence {x n(i) } such that {xn(i) } ∩ W = ∅.
Then since yn(i) ∈ gn(i) (xn(i) ) ⊂ gi (xn(i) )(i ≥ 1) and {yn(i) } −→ y, {xn(i) }contains a subsequence which converges to y. This contradiction implies
that {xn } −→ y.
(3): Since X is q and wcc, by Theorem 2.3 and (2) of this theorem, there
is a g-function g such that, whenever y n ∈ gn (xn ) (n ≥ 1) and {yn } −→ y,then {xn } −→ y, and {gn (x)} is a neighbourhood base of x. To see that g
is a Nagata function, let yn ∈ gn (x) ∩ gn (xn ) (n ≥ 1). Then {yn } −→ x.
Hence {xn } −→ x.
(4): Let g be a cs-stratifiable wθ-function of X. Then g is a q-function.
Indeed, let xn ∈ gn (x) (n ≥ 1), then {xn } has a cluster point since x ∈gn (x), xn , x ∈ gn (x) (n ≥ 1). Therefore, {gn (x)} is a neighbourhood base of
x by Theorem 2.3. Now, suppose that y n ∈ gn (p) and xn , p ∈ gn (yn ) (n ≥ 1).
Then {xn } has a cluster point x and {yn } converges to p. If x 6= p, thenK-SEMIMETRIZABILITIES AND C-STRATIFIABILITIES185x∈/ {yn |n ≥ m} ∪ {p} = C for some m ∈ N. Therefore x ∈/ g k (C) for somek ≥ m. This contradiction implies x = p, and hence g is a θ-function.
(5): Let g be a cs-stratifiable, wγ-function of a space X. Suppose thatyn ∈ gn (p), xn ∈ gn (yn ) (n ≥ 1). Then {xn } has a cluster point x, and{yn } converges to p since g is a q-function. If p 6= x, then x ∈/ {y n |n ≥m} ∪ {p} = C for some m ∈ N. Therefore x ∈/ g k (C) for some k ≥ m, which
is a contradiction.
(6): Let g be a cs-stratifiable w∆-function of X. Since g is a β-function,g is a semistratifiable function. Now, suppose that x n , p ∈ gn (yn ) (n ≥ 1).
Then {yn } converges to p and {xn } has a cluster point x. If x 6= p, thenx ∈/ {yn |n ≥ m} ∪ {p} = C for some m ∈ N. Hence x ∈/ g k (C) for somek ≥ m. This contradiction implies that the function g satisfies condition (K).
Remark 2.11. (1) In the class of strongly α-spaces, the implications (3)-(6)in Theorem 2.10 are true by Theorem 2.4, (1) follows from [17; Theorem
5.2] and (2) follows from [31; Proposition 4.7].
(2) In the class of weak c-stratifiable regular spaces, the implications (3)(6) in Theorem 2.10 are true by Theorem 2.3. For the implications (1) and(2), let g be a g-function satisfying the respective conditions. Then, since∩n≥1 gn (x) = {x}, (1) and (2) are also true by a similar argument to theproof of Theorem 2.10.
The following questions regarding c-stratifiable spaces and strongly αspaces are natural.
Question 2.12. When are c-stratifiability and strong α-ness coincident ?
Question 2.13. Is every paracompact first coutable c-stratifiable spacestrongly α ?

3. K-semimetrizable spaces
Definition 3.1. Let X be a space. Then a function d : X × X −→ R
is called a semimetric if (i) d(x, y) ≥ 0, (ii) d(x, y) = 0 ⇐⇒ x = y and(iii) d(x, y) = d(y, x). X is called a semimetrizable space or X has acompatible semimetric if there exists a semimetric d on X such that forany subset M ⊂ X, x ∈ M ⇐⇒ d(x, M ) = 0, or equivalently, for any
x ∈ X and any open neighbourhood U of x, x ∈ intB(x; ) ⊂ B(x; ) ⊂ U
for some >0; where B(A; δ) = {y ∈ X|d(A, y) =inf{d(a, y)|a ∈ A}<δ}
for each δ>0 and any subset A ⊂ X and B(x; δ) = B({x}; δ). Then, fora sequence {xn } in a semimetrizable space (X, d), lim n→∞ d(x, xn ) = 0

186

I. YOSHIOKA⇐⇒ {xn } −→ x in X. A semimetrizable space X with a compatible semimetric d is K-semimetrizable [26] if d(H, K) =inf{d(x, y)|x ∈ H, y ∈ K}>0
for any disjoint compact subsets H and K. In this situation, d is called aK-semimetric on X.
It is well known [8] that a space X is semimetrizable if, and only if, it isa first countable, semistratifiable space.
Definition 3.2. Let (X, d) be a semimetrizable space. For each n ∈ N,
we put Gn = {intB(x; )|δB(x; )<1/n}, where for subset A of X, δA =sup{d(x, y)|x, y ∈ A}. The semimetric d is said to be full if G n is a coverof X for each n ∈ N, or equivalently, if d satisfies Arhangel’ski ĭ’s condition(AN): At each point, there is a neighbourhood of arbitrarily small diameter[1]. A space X is called full K-semimetrizable if X has a compatible fullK-semimetric.
Zenor investigated spaces with a regular G δ -diagonal and gave the following result.
Theorem 3.3 ([32; Theorem 2]). For a space X, the following conditions
are equivalent.
(1) X has a development satisfying the 3-link property.
(2) X is a w∆-space with a regular Gδ -diagonal.
(3) X has a compatible semimetric d satisfying(I) If {xn } −→ x and {yn } −→ x, then lim d(xn , yn ) = 0, andn→∞(II) If {xn } −→ x, {yn } −→ y and x 6= y, then there exist r>0 andm ∈ N such that d(xn , yn )>r for each n ≥ m.
In substance, the first part of the following theorem is proved in (1)⇐⇒(3)of [32; Theorem 2] or [22; Lemma 5.3].
Theorem 3.4. (1) For a space X, the following conditions are equivalent.
(i) X is a developable space.
(ii) X has a compatible full semimetric d.
(iii) X has a compatible semimetric d satisfying (I) of Theorem 3.3.
(2) A space X is developable T2 if, and only if, it is wθ, β and has a∗Gδ -diagonal.

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Proof of (2). We only prove the “if” part. Every β-space with a G ∗δ diagonal is semistratifiable [17; Theorem 5.2] and every semistratifiable wθspace is w∆ [18; Proposition 4.5]. Hence X is developable [17; Theorem 2.5].
For a semimetric space, we have the following characterization. A regularspace X is semimetrizable if, and only if, it is a q, β-space with a G ∗δ -diagonal.
Indeed, let g be a q-function and {Gn } be a G∗δ -diagonal sequence of aspace X. We put hn (x) = gn (x) ∩ st(x, Gn ), then {hn (x)} is a neighbourhood base of x. Also, X is semistratifiable from the proof of Theorem 3.4(2).
For the converse implication, see [17].
The following theorem improves the result [11; Proposition 2.7] or [22;Theorem 5.2] that a space X is K-semimetrizable if, and only if, it is cstratifiable and semimetrizable.
Theorem 3.5. For a space X, the following conditions are equivalent.
(1) X is a K-semimetrizable space.
(2) X is a c-stratifiable semimetrizable space.
(3) X is a cs-stratifiable q, β-space.
(4) X has a compatible semimetric d satisfying (II) of Theorem 3.3.
(5) X has a compatible semimetric d such that, x ∈/ B(K; 1/m) for somem ∈ N, whenever x ∈/ K and K is compact.
Proof. (1)=⇒(2) is proved in [22; Theorem 5.2] and (2)=⇒(3) is evident.
(3)=⇒(1): Let g be a cs-stratifiable q, β-function of X. Then by Theorems 2.3 and 2.10, g is a c-stratifiable and semistratifiable function, and{gn (x)} is an open neighbourhood base of x for every x ∈ X. Now, we defined(x, x) = 0 and d(x, y) =1/inf{j|x ∈/ g j (y) and y ∈/ gj (x)} if x 6= y. By [22;Theorem 5.2], (X, d) is K-semimetrizable.
(1)=⇒(4): Let d be a compatible K-semimetric on X. Suppose that{xn } −→ x, {yn } −→ y and x 6= y. Since X is T2 , for some m ∈ N,
H = {xn |n ≥ m} ∪ {x} and K = {yn |n ≥ m} ∪ {y} are disjoint compact
subsets. Therefore we have that 0<d(H, K) ≤inf{d(x n , yn )|n ≥ m}.
(4)=⇒(5): Suppose that x ∈/ K, where K is compact, and x ∈ B(K; 1/n)
for each n ∈ N with respect to the semimetric d satisfying the condition of(4). Then there exists a sequence {z n } such thatzn ∈ B(K; 1/n) ∩ intB(x; 1/n).
Hence {zn } −→ x. Also d(xn , zn )<1/n for some sequence {xn } ⊂ K.
Then there exist subsequences {xn(i) } ⊂ {xn } and {zn(i) } ⊂ {zn } suchthat {xn(i) } −→ p for some p ∈ K and {zn(i) } −→ x. Therefore there exist

188

I. YOSHIOKAj, m ∈ N such that d(xn(i) , zn(i) ) ≥ 1/m for each i ≥ j. On the other hand,d(xn(k) , zn(k) )<1/n(k) for some n(k) ≥ max{n(j), m}, which is a contradiction.
(5)=⇒(1): Let d be a compatible semimetric satisfying the condition of(5). If H and K are disjoint compact subsets of X with d(H, K) = 0,then lim d(xn , yn ) = 0 for some sequences {xn } ⊂ H, {yn } ⊂ K. Since X
is first countable, there exist subsequences {x n(i) } ⊂ {xn }, {yn(i) } ⊂ {yn }and points x ∈ H, y ∈ K satisfying {xn(i) } −→ x, {yn(i) } −→ y. Sincey ∈/ B(H; 1/m) for some m ∈ N, we have that B(y; 1/k) ∩ B(H; 1/k) = ∅for some k ≥ m. This contradicts the fact that d(x n(i) , yn(i) )<1/k andd(y, yn(i) )<1/k for some n(i) ∈ N.
Remark 3.6. (1) The space Y in Example 4.9 is c-stratifiable β, but notq, and the Sorgenfrey line is c-stratifiable q, but not β.
(2) The space X in Example 4.6 is Moore (hence, X has a G ∗δ -diagonal),
but not K-semimetrizable, and the Nagata space X in Example 4.9 is Ksemimetrizable, but not Moore.
(3) The space Y in Example 4.9 is stratifiable (hence c-stratifiable) Fréchetas the perfect image of a Nagata space (hence, K-semimetrizable), but Y isnot semimetrizable (not even q).
Proposition 3.7. Every K-semimetrizable space has a G ∗δ -diagonal.
Proof. By Theorems 2.3 and 3.5, let g be a cs-stratifiable q, β-functionof X such that {gn (x)} is a neighbourhood base of x. For each n ∈ N,
we put Gn = {gn (x)|x ∈ X}. To see that the sequence {G n } is a G∗δ diagonal, suppose that x 6= y ∈ ∩n≥1 st(x, Gn ). Then there exist zn ∈gn (y) ∩ st(x, Gn ) (n ≥ 1). Hence {zn } −→ y and x, zn ∈ gn (xn ) for somesequence {xn }. Then {xn } −→ x and y ∈/ C = {xn |n ≥ m} ∪ {x} for somem ∈ N. Hence y ∈/ gk (C) for some k ≥ m. This is a contradiction.
The following theorem gives a condition for strong α-ness andc-stratifiability to be equivalent, and follows directly from Theorems 2.4,
2.8 and 3.5 and Proposition 3.7.
Theorem 3.8. For an orthocompact β, q-space, the following conditions
are equivalent.
(1) X is K-semimetrizable.
(2) X has a G∗δ -diagonal.
(3) X is strongly α.

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189(4) X is cs-stratifiable.
An analogue to Theorem 3.5 for the class of regular spaces follows directlyfrom Theorem 2.3.
Theorem 3.9. For a regular space X, X is K-semimetrizable if, and only
if, it is weak c-stratifiable q, β.
We next give some partial answers to the question of Burke [6; Question2] on what minimal topological condition on a Moore space (or semimetricspace) will ensure that the space is K-semimetrizable.
Theorem 3.10. (1) Every T2 , orthocompact developable space X is Ksemimetrizable.
(2) Every regular orthocompact semistratifiable q-space (hence, regularorthocompact semimetrizable space) X is K-semimetrizable.
(3) Every regular orthocompact c-semistratifiable q, β-space X is K-semimetrizable.
(4) Every regular k-semistratifiable q-space X is K-semimetrizable.
Proof. Since a developable T2 -space has a Gδ (2)-diagonal, (1) follows
from Theorems 2.8 and 3.5. Since every semistratifiable T 2 -space has aGδ -diagonal, (2) follows from Theorems 2.8 and 3.5. For (3), since X issemistratifiable, (3) follows from (2). (4) follows from Theorems 2.3, 2.4
and 3.5.
Remark 3.11. (1) With regards to (2) of Theorem 3.10, it is already known[1; page 133] or [22; page 441], that every paracompact semimetrizable spaceis K-semimetrizable.
(2) In (2) and (3) of Theorem 3.10, we can not change orthocompactness
to subparacompactness by Example 4.6.
(3) In (4) of Theorem 3.10, we already know that a space is regular ksemistratifiable q if, and only if, it is Nagata [31; Theorem 2.1]. But, we do
not know whether every T2 , k-semistratifiable q-space is c-stratifiable. (Ifthis answer is affirmative, then every T 2 , k-semistratifiable q-space is first
countable and Nagata.) The converse of (4) does not hold, because the space
Ψ in Example 4.5 is not k-semistratifiable.
In the following theorem, the equivalence of (1) and (4) is proved in [22;Theorem 5.4].

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Theorem 3.12. For a space X, conditions (1)-(5) are all equivalent and(5) =⇒ (6) holds.
(1) X is a full K-semimetrizable space.
(2) X has a development {Gn } such that if K1 and K2 are disjoint compact
subsets, then st(K1 , Gm ) ∩ K2 = ∅ for some m ∈ N.
(3) X has a development {Gn } such that if p ∈/ C, where C is the union of aconvergent sequence and any one point of its limit points, then p ∈/ st(C, G m )for some m ∈ N.
(4) X satisfies one of the equivalent conditions in Theorem 3.3.
(5) X is a wθ, β-space with a regular G δ -diagonal.
(6) X is a developable c-stratifiable space.
Proof. (1)=⇒(2): Let d be a compatible full K-semimetric on X. For eachn ∈ N, we put Gn = {intB(x; )|δB(x; )<1/n}. Then {G n } is a developmentof X since d is a full semimetric. For, suppose that x ∈ X and x n ∈st(x, Gn ) \ U (n ≥ 1) for some open neighbourhood U of x. Then x, x n ∈ Gn
and δGn <1/n for some Gn ∈ Gn , which is a contradiction. Now, supposethat K1 and K2 are disjoint compact subsets and xn ∈ st(K1 , Gn ) ∩ K2 foreach n ∈ N. Then yn ∈ Gn ∩ K1 and xn ∈ Gn for some Gn ∈ Gn . SinceδGn <1/n (n ≥ 1), lim d(xn , yn ) = 0. This contradicts d(K1 , K2 )>0.
n→∞(2)=⇒(3): Let {Gn } be a development of X satisfying (2). To see that X is
T2 . let x 6= y and xn ∈ st(x, Gn ) ∩ st(y, Gn ) for each n ∈ N. Then {xn } −→ x
and {xn } −→ y. Given any open neighbourhood U of x with y ∈/ U, S ={xn |n ≥ m} ∪ {x} ⊂ U for some m ∈ N. Then st(y, G k ) ∩ S = ∅ for somek ≥ m. This contradicts {xn } −→ y. Next, suppose that p ∈/ K, where K iscompact, and p ∈ ∩n≥1 st(K, Gn ). Then an ∈ st(p, Gn ) ∩ st(K, Gn )(n ≥ 1).
Hence an ∈ st(xn , Gn ) for some sequence {xn } in a sequentially compact K,and {xn } contains a subsequence {xn(i) } converging to some point x ∈ K.
Since X is T2 , L = {xn(i) |n(i) ≥ m} ∪ {x} and H = {an(i) |n(i) ≥ m} ∪ {p}are disjoint for some m ∈ N. Therefore, a n(k) ∈ st(L, Gn(k) ) ∩ H = ∅ forsome n(k) ≥ m, which leads to a contradiction.
(3)=⇒(4): Let {Gn } be a development of X such that Gn+1 is a refinementof Gn and satisfies (3). We now show that {G n } satisfies the 3-link property.
Suppose that x 6= y and for each n ∈ N, there exists G n ∈ Gn such that xn ∈Gn ∩ st(x, Gn ) and yn ∈ Gn ∩ st(y, Gn ). Since {xn } −→ x, {yn } −→ y and
X is T2 , y ∈/ C = {xn |n ≥ m} ∪ {x} for some m ∈ N. Hence y ∈/ st(C, G k )for some k ≥ m. Then yl ∈ X \ st(C, Gk ) for some l ≥ k and xl ∈ C.
Therefore, yl ∈ Gl ⊂ st(xl , Gl ) ⊂ st(C, Gk ), which is a contradiction.
(4)=⇒(5): Let X be a w∆-space with a regular G δ -diagonal. Then X
satisfies condition (5).

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191(5)=⇒(1): By Theorem 3.4, X is a developable space with a regular G δ diagonal. Hence there exists a compatible semimetric d on X satisfying (I)and (II) of (3) in Theorem 3.3. Then d is full by (3)=⇒(1) of [32; Theorem
2]. To see that d is K-semimetric, suppose that d(K, H) = 0 for some disjoint compact subsets K and H. Then there are sequences {x n } ⊂ K and{yn } ⊂ H such that lim d(xn , yn ) = 0. On the other hand, since X is an→∞q-space with a G∗δ -diagonal, X is first countable. Hence {x n } ({yn }) contains a subsequence {xn(i) } ({yn(i) }) converging to a point x ∈ K (y ∈ H,
respectively). Hence there are k, m ∈ N such that d(x n(i) , yn(i) ) ≥ 1/m foreach i ≥ k by (II). This is a contradiction. Finally, (5)=⇒(6) follows from
Theorems 2.8 and 3.4.
Remark 3.13. (1) The space Ψ in Example 4.5 is Moore and K-semi-metrizable, but not full K-semimetrizable.
(2) Every w∆-space is wθ and β. Although the converse is an open problem [18; Problem 4.10], (4) ⇐⇒ (5) of Theorem 3.12 (or (2) of Theorem 3.4)
may be a slight progress to [32; Theorem 2] ([17; Theorem 2.5], respectively).
(3) The space X in Example 4.8 is T2 metacompact, full K-semi-metrizable, but not regular.
Question 3.14. Is every normal metacompact, full K-semimetrizable space,
metrizable?
We next investigate conditions for spaces to be developable and K-semimetrizable.
Theorem 3.15. Consider the following conditions for a space X.
(1) X is developable and K-semimetrizable.
(2) X is K-semimetrizable wθ.
(3) X is cs-stratifiable wθ and β.
(4) X is strongly α, wθ and β.
(5) X is developable T2 .
T hen, (1), (2) and (3) are equivalent.
Moreover, if X is orthocompact, then all conditions are equivalent.
Proof : (1)⇒(2) ⇒(3) are evident. For (3)=⇒(1), X is K-semimetrizableby Theorem 3.5. Since X is semistratifiable and θ by Theorem 2.10, X
is developable [18; Remark 4.8]. (4)=⇒(3) follows from Theorem 2.4, and(3)=⇒(5) is evident. Moreover, if X is orthocompact, (5)=⇒(4) follows from
Theorem 2.8.

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Martin [26] showed that a locally connected rim-compact T 2 -space X is
K-semimetrizable if, and only if, it is developable γ.
Definition 3.16. A space X is said to be rim-compact if each point of X
has a neighbourhood base consisting of open subsets with compact boundaries. A space X is locally connected if each point of X has a neighbourhoodbase consisting of connected open subsets.
We need the following lemma.
Lemma 3.17. (1) Every locally connected rim-compact weak c-stratifiable(or, cs-stratifiable) space X is a c-stratifiable γ-space.
(2) Every pseudocompact Tychonoff weak c-stratifiable (or, cs-stratifiable)
space X is a c-stratifiable γ-space.
Proof. (1): First, let g be a weak c-stratifiable function of X. Then, we
can assume that gn (x) is connected for every x ∈ X and each n ∈ N. To
see that X is a γ-space, we use the same method given in the proof of [26;Theorem 4]. Suppose that K ⊂ W , where K is non-empty compact and W
is open. Then there is an open subset G such that K ⊂ G ⊂ W and theboundary ∂G of G is compact. Since K ∩ ∂G = ∅, g m (K) ∩ ∂G = ∅ for somem ∈ N. Let K = ∪{Kα |α ∈ A}, where Kα is a connected component of
K. Since gm (Kα ) is connected for each α ∈ A, gm (K) = ∪α∈A gm (Kα ) ⊂ G.
Hence g is a γ-function by [23; Theorem 2.1]. Since X is first countable, g isa c-stratifiable function by [22; Theorem 1.3]. Next, let g be a cs-stratifiable
function of X. To see that {gn (x)} is a neighbourhood base of x for everyx ∈ X, in the above proof, let K be a single point x. Since {x} ∩ ∂G = ∅and ∂G is compact, we have that gm (x) ∩ ∂G = ∅ for some m ∈ N. This
asserts that gm (x) ⊂ G, which implies that X is first countable and regular.
Therefore X is c-stratifiable by Theorem 2.3, and hence X is a γ-space.
(2): Let g be a weak c-stratifiable function or a cs-stratifiable function of X. By regularity of X, we assume that g n+1 (x) ⊂ gn (x). Since∩n≥1 gn (x) = {x}, X is first countable by [27; Lemma 2.3]. Hence X is
c-stratifiable by Theorem 2.3 and hence, X is γ by [22; Theorem 4.2].
Theorem 3.18. Let X be a locally connected rim-compact space or a pseudocompact Tychonoff space. Then the following conditions are equivalent.
(1) X is developable and K-semimetrizable.
(2) X is K-semimetrizable.
(3) X is T2 , developable and γ.
(4) X is weak c-stratifiable and β.

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193(5) X is cs-stratifiable and β.
(6) X is T2 , γ and β.
Proof. First, we note that every γ, β-space is developable [18; Proposition
4.2]. (1)⇐⇒(4) and (1)⇐⇒(5) follow from Theorem 3.5 and Lemma 3.17.
(1)=⇒(2)=⇒(4) and (1)=⇒(3)=⇒(6) is evident. Since every T 2 , γ-spaceis c-stratifiable, (6)=⇒(5) is true.
By the proof of the above theorem and Theorem 3.5, we have that in theclass of T2 , γ-spaces, the following properties are coincident: (1) developable
and K-semimetrizable, (2) K-semimetrizable, (3) developable and (4) β.
The next theorem follows from Theorem 3.10.
Theorem 3.19. For an orthocompact T 2 -space X, X is developable and Ksemimetrizable if, and only if, it is developable
A Tychonoff space X is called a p-space [2] if in the Stone- C̆ech compactification βX, there is a sequence {G n } of open covers of X such that∩n≥1 st(x, Gn ) ⊂ X for every x ∈ X. Every locally compact T 2 -space is a
p-space.
Burke [5] showed that there is a locally compact T 2 -space with a Gδ diagonal, which is not w∆. But, it is known that every locally compact
semistratifiable T2 -space or every θ-refinable p-space with a G δ -diagonal is
Moore [8, 21]. Then we have the following result by Theorem 3.10.
Theorem 3.20. For a metacompact p-space X, X is Moore and K-semi-metrizable if, and only if, it has a G δ -diagonal.
The next result was studied by Kotake [20] in the class of regular spaces.
Theorem 3.21. For a space X, the following conditions are equivalent.
(1) X is Nagata.
(2) X is K-semimetrizable wcc.
(3) X is cs-stratifiable wN .
(4) X is strongly α, wN .
(5) X is a wN-space with a G∗δ -diagonal.
(6) X is regular semimetrizable wcc.
Proof. Every Nagata space is stratifiable and first countable, hence it
is c-stratifiable q and β. Therefore (1)=⇒(2) and (2)=⇒(3) follow from

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Proposition 1.4 and Theorem 3.5, and (3)=⇒(1) follows from Theorem 2.10.
(1)=⇒(4) and (4)=⇒(3) follow from Theorems 2.4 and 2.8. Also, (1)=⇒(5)is evident. To prove (5)=⇒(4), let g be a wN -function and {G n } be a G∗δ diagonal sequence. Since regularity is not used to show that every β-spacewith a G∗δ -diagonal is semistratifiable [17; Theorem 5.2], X is a subparacompact wN -space. Then X is metacompact by [18; Corollary 3.5]. Hence X isstrongly α by Theorem 2.8. (1)=⇒(6) is evident. Finally, since every regular semistratifiable space has a G∗δ -diagonal [14; Theorem 5.11], (6)=⇒(5)follows from Proposition 1.4.
Regarding Question 2.12, we have the following corollary which follows
from the fact that every wcc-space is β.
Corollary 3.22. For a wN -space, the classes of the following spaces are all
coincident.
(1) Nagata spaces, (2) strongly α-spaces, (3) c-stratifiable spaces, (4) Ksemimetrizable spaces and (5) spaces with a G ∗δ -diagonal.
Remark 3.23. Ceder [7; page 114] asked whether every paracompact
semimetrizable space must be a Nagata space. Heath [16] showed that there
is a paracompact K-semimetrizable cosmic (the continuous image of a separable metric space) space which is not a stratifiable space (hence, neither
k-semistratifiable [24; Example 4.2] nor wcc). He also posed the following problem: What topological condition is necessary for a paracompact
semimetrizable (= K-semimetrizable) space to be an M 3 -space? As a remark to this problem, one can note that in the class of regular semimetrizable
spaces, Nagata spaces, k-semistratifiable spaces and wcc-spaces are coincident.
4. Metrizabilities and examples
We begin this section with metrizations of wM -spaces. The concept ofwM -spaces was given by Ishii [19]. Here we define a wM -space by an equivalent condition given by Hodel.
Definition 4.1 [18; Theorem 5.2]. A space X is wM if, and only if, it iswγ and wN .
The following implications are well known.
An M -space (in the sense of Morita) =⇒ a wM -space =⇒ a w∆-space.
The class of wM -spaces is contained in the class of wθ, wcc-spaces. Therefore, we consider metrizations for the class of wθ, wcc-spaces. Metrizations

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195

for this class was studied in [28]. For metrizations of wM -spaces, Martin[25] proved that every regular c-semistratifiable wM -space is metrizable, andIshii [19] proved that every normal wM -space with a G ∗δ -diagonal is metrizable. On the other hand, the space Ψ in Example 4.4 is a c-stratifiable
Moore γ-space which is not metrizable.
Theorem 4.2. Let X be a wθ, wcc-space. Then X is metrizable if X satisfies
any one of the following statements.
(1) X is K-semimetrizable.
(2) X is strongly α.
(3) X is cs-stratifiable.
(4) X has a G∗δ -diagonal.
(5) X is regular c-semistratifiable.
Proof. For all conditions (1)-(5), X is a wN -space by Proposition 1.4.
Hence for (1)-(4), X is a wθ, Nagata space by Theorem 3.21. Therefore,
X is metrizable [30; Theorem 5]. For (5), since every wcc-space is β, X
is regular c-semistratifiable β, hence X is semistratifiable. Then X is wcc
Moore [18; Corollary 4.6], which implies that X is metrizable [31; Corollary
3.6].
Remark 4.3. In Theorem 4.2, the condition wθ (wcc) can not be weakened
to q (β, respectively). Indeed, the Nagata-space X in Example 4.9 is a q,
wcc-space which satisfies all of the conditions (1)-(5) in Theorem 4.2, but
is not metrizable. Also, the space Ψ in Example 4.5 is a γ, β-space which
satisfies all of the conditions (1)-(5) in Theorem 4.2, but is not metrizable.
The second part (2) of the next theorem is a generarization of Lee’s result[22] that every pseudocompact Tychonoff stratifiable space is metrizable.
Theorem 4.4. (1) Every locally connected rim-compact k-semistratifiable
space X is metrizable.
(2) Every pseudocompact Tychonoff k-semistratifiable space X is metrizable.
Proof. First, we show that if X satisfies the conditions of (1), then X isa first countable T2 -space. Let g be a k-semistratifiable function such thatgn (x) is connected. To see that {gn (x)} is a neighbourhood base of x forevery x ∈ X, suppose that x ∈ U and gn (x) \ U 6= ∅ (n ≥ 1), where U
is open. Then there is an open neighbourhood W of x such that W ⊂ U
and the boundary ∂W is compact. Since g m (x) ∩ ∂W = ∅ for some m ∈ N,

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gm (x) = (gm (x) ∩ W ) ∪ (gm (x) \ W ) is not connected. This contradiction implies that {gn (x)} is a neighbourhood base of x. To see that X is Hausdorff,
let x 6= y and xn ∈ gn (x) ∩ gn (y) (n ≥ 1). Then for any open neighbourhood U of x with y ∈/ U , K = {xn |n ≥ m} ∪ {x} ⊂ U for some m ∈ N.
Hence gl (y) ∩ K = ∅ for some l ≥ m, which is a contradiction. Next, inboth cases, X is a γ-space by Theorem 2.4 and Lemma 3.17. Also, X is a
wcc-space. Indeed, let g be a k-semistratifiable function such that, whenever bn ∈ gn (an ) (n ≥ 1) and {bn } −→ b, then {an } −→ b. Now, supposethat yn ∈ gn (xn ) (n ≥ 1) and {yn } has a cluster point y. Since X is first
countable, there exists a subsequence {y n(i) } of {yn } converging to y andyn(i) ∈ gi (xn(i) ) (n ≥ 1). Hence {xn(i) } converges to y, which implies that g
is a wcc-function. Finally, every γ, wcc T 2 -space is metrizable [31; Corollary
3.6].
We note that Martin [26; Example 3] showed that there exists a locallyconnected locally compact K-semimetrizable Moore space X which is notnormal. This space is not wcc by Theorem 3.21.
As regards to Theorem 4.4, (2) is proved in [30; Corollary 4] in a different
way, and as for (1), every locally compact T 2 (even sieve-complete regular)
k-semistratifiable is metrizable [30; Theorem 18].
Example 4.5. [22; Example 6.6] The space Ψ in [13; 5I] is Moore andK-semimetrizable that is not full K-semimetrizable. First, it is known
that Ψ is a locally compact pseudocompact separable Moore c-stratifiable
space that is not metacompact. To see that Ψ is orthocompact, for any
E
E = {xE
k |k ∈ N} ∈ E, where {xk |k ∈ N} is an infinite subsequence of N, weE
put B(ωE , n) = {ωE } ∪ {xE
n , xn+1 , ...}(n ∈ N). Then any open cover G of
Ψ has the refinement H = {{n}|n ∈ N} ∪ {B(ω E , n(E))|E ∈ E}, where for
any E ∈ E, B(ωE , n(E)) ⊂ G for some G ∈ G and some n(E) ∈ N. And∩W is open for any W ⊂ H. Therefore, Ψ is strongly α by Theorem 2.8.
Then Ψ is K-semimetrizable and γ by Theorem 3.5 and Lemma 3.17. ButΨ does not have a regular Gδ -diagonal [27; Theorem 2.6], and not wcc from
Theorem 3.21. Hence it is not full K-semimetrizable by Theorem 3.12 andnot k-semistratifiable since every first countable k-semistratifiable space is
wcc.
Example 4.6. [6] Burke constructed the separable Moore (hence, semimetrizable) space X which is not K-semimetrizable. Hence, X is a csemistratifiable α-space which is neither strongly α nor cs-stratifiable byTheorems 2.4 and 3.5. Also, X is not metacompact by Theorem 2.8 and not

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197

γ.
Example 4.7. [18; Example 4.14]. The Sorgenfrey line K is a paracompactγ-space with a Gδ -diagonal. Hence K is strongly α and c-stratifiable, but
not semistratifiable (not even β [18; Proposition 4.2]).
Example 4.8. [9: Example 5.3.4] There exists a metacompact full Ksemimetrizable space which is neither wcc nor regular.
Indeed, let X be the space of real numbers with the topology generated
by the neighbourhood system {U(x)|x ∈ X}, where U(x) = {U n (x)|n ∈ N}
and((x − 1/n, x + 1/n)
if x 6= 0,
Un (x) =(x − 1/n, x + 1/n) \ {1/k|k ∈ Z \ {0}} if x = 0,
where Z denotes the set of integers. It is well known that X is a metacompact
T2 -space which is not regular. For each x ∈ X, we put(
Un (x) \ {0} if x 6= 0,
Wn (x) =Un (x)
if x = 0.
Let Wn = {Wn (x)|x ∈ X} for each n ∈ N. Then it is easily seen that thesequence {Wn } is a development satisfying the 3-link property. Therefore,
X is full K-semimetrizable. Then X is strongly α and c-stratifiable by Theorem 2.8. Also, if X is wcc, then it is metrizable by Theorem 4.2, which isa contradiction.
Example 4.9. [24; Example 4.3] There exist a first countable stratifiable
space X and a perfect map f from X onto a non-q-space Y . Then X is a
Nagata space (hence, X is K-semimetrizable) which is not wθ [30; Theorem
5] and Y is a stratifiable space which is not q. Then, Y is strongly α andc-stratifiable but not semimetrizable.
Example 4.10. [10; Example 4.2] A regular full K-semimetrizable spacethat is not orthocompact. Let R = {(x, y)|x, y are rational and y>0}. Let
J be the set of irrational numbers and let X = R ∪ (J × {0}). We give
R the usual subspace topology T ∗ . For each x ∈ J and each >0, letB(x, ) = {(x, 0)} ∪ {(x + k, h)||k|<h<}. Then T ∗ ∪ {B(x, )|x ∈ J, >0} isa basis for a topology on X. Then X is a separable Moore space that is notorthocompact. Also, X has a development satisfying the 3-link property,
hence full K-semimetrizable and c-stratifiable.
But I don’t know whether this space is strongly α.

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I. YOSHIOKA

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Iwao Yoshioka
139 Mitsuyoshi, Kouryo-cho, Kita-Katsuragi, Nara 635-0823, Japane-mail address: yoshioka iwao@ybb.ne.jp(Received March 6, 2007 )(Revised June 5, 2007 )