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ADVANCES IN MATHEMATICS
Vol.42, No.6
Dec., 2013
A Note on Weak Stratifiable Spaces
YANG Erguang∗ , WU Dailong∗∗
(School of Mathematics and Physics, Anhui University of Technology, Maanshan, Anhui, 243002,
P. R. China)
Abstract: We show that a space with a countable closed k-network does not need to be
a weak stratifiable space which partially answers a question posed by Lin in [The properties
of k-semi-stratifiable spaces and related questions, Advances in Mathematics(China), 2012,
41(1): 1-6]. Moreover, we present a sufficient condition for a topological space to be a weak
stratifiable space.
Key words: weak stratifiable spaces; weak MCP spaces; pair sk-network
MR(2000) Subject Classification: 54E35; 54E99 / CLC number: O189.1
Document code: A
Article ID: 1000-0917(2013)06-0885-04
0 Introduction
The notion of a weak stratifiable space was introduced by Peng[6] to generalize stratifiable
spaces. In [7], weak stratifiable spaces were called contraconvergent spaces. In [6], several questions concerning the interrelations between the weak stratifiable spaces and some related spaces
were posed. One of these questions is as follows: Is every weak stratifiable space submesocompact? Since every space with a σ-closure-preserving k-network is submesocompact, Lin posed
the following question[5] : What is the interrelation between the weak stratifiable spaces and the
spaces that have σ-closure-preserving k-networks? In this note, we shall give a partial answer
to this question. Moreover, we show that a space with a σ-cushioned pair sk-network is weak
stratifiable.
Let X be a space. τ denotes the topology of X and A◦ denotes the interior of A. The set
of all positive integers is denoted by N while xn denotes a sequence.
A collection P of subsets of X is called a k-network if for any compact subset K and
open set U with K ⊂ U , there exists a finite subcollection P of P such that K ⊂ P ⊂
U . A collection P of pairs of subsets of X is called a pair sk-network if P1 ⊂ P2 for each
(P1 , P2 ) ∈ P and whenever x ∈ U ∈ τ , there exists a finite subcollection P of P such that
x ∈ ( {P1 : (P1 , P2 ) ∈ P })◦ ⊂ {P2 : (P1 , P2 ) ∈ P } ⊂ U , while P is called cushioned if
{P1 : (P1 , P2 ) ∈ P } ⊂ {P2 : (P1 , P2 ) ∈ P } for any P ⊂ P.
A g-function for a space X is a map g : N × X → τ such that for every x ∈ X and n ∈ N,
x ∈ g(n, x) and g(n + 1, x) ⊂ g(n, x).
Received date: 2011-10-17. Revised date: 2012-04-15.
Foundation item: This work is supported by NSFC (No. 11001001, No. 10971092) and Foundation of Anhui
Educational Committee for Excellent College Young Scholars (No. 2010SQRL041).
E-mail: ∗ egyang@126.com; ∗∗ wdl18955532992@126.com
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X is called a weak stratifiable space[6] if there is a g-function g for X such that if x is a
cluster point of yn and yn ∈ g(n, xn ) for all n ∈ N, then x is a cluster point of xn . X is called
a weak MCP space[6] if there is a g-function g for X such that if yn has a cluster point and
yn ∈ g(n, xn ) for all n ∈ N, then xn has a cluster point.
In [7], weak stratifiable spaces were called contraconvergent spaces and weak MCP spaces
were called weak contraconvergent (wcc) spaces in [9].
All spaces are assumed to be Hausdorff unless otherwise stated.
1 Main Results
Question 1.1[5] What is the interrelation between the weak stratifiable spaces and the
spaces that have σ-closure-preserving k-networks?
The following example gives a partial answer to the above question.
Example 1.2 There exists a non-weak-stratifiable space which has a countable closed
k-network.
Proof The space X used here is a modification of the Arens space and is a reformulation
of the space X of Example 1 in [10]. Let X = {0} ∪ N ∪ (N × N) and let F be the collection
of all finite subsets of N. Denote by NN the set of all maps from N to N and A(n, m) =
{(n, k) : k ≥ m}, B(n, m) = {n} ∪ A(n, m) for each n ∈ N. For each f ∈ NN and F ∈ F , put
H(F, f ) = {A(n, f (n)) : n ∈ N \ F }. For each x ∈ X, the neighborhood base B(x) of x is
defined as follows:
⎧
⎨{{x}},
B(x) = {B(x, m) : m ∈ N},
⎩
{{x} ∪ H(F, f ) : F ∈ F , f ∈ NN },
x ∈ N × N;
x ∈ N;
x = 0.
That X is not a weak-stratifiable space has been shown in [10]. We shall show that X has a
countable closed k-network.
Let K be a compact subset of X. The following Claims 1 and 3 are clear.
Claim 1 K ∩ N is finite.
Claim 2 The set {n ∈ N : K ∩ A(n, 1) = ∅} is finite.
Proof of Claim 2 Suppose that K ∩ A(nk , 1) = ∅ for infinitely many nk . For each k ∈ N,
choose xnk ∈ K ∩ A(nk , 1) and denote xnk = (nk , mk ). Let F = ∅ and f ∈ NN be such that
mk + 1,
n = nk , k ∈ N;
f (n) =
1,
otherwise.
Then U = {0} ∪ H(F, f ) is an open neighborhood of 0 such that U ∩ {xnk : k ∈ N} = ∅, which
implies that 0 is not an accumulation point of {xnk : k ∈ N}. Thus {xnk : k ∈ N} has no
accumulation point, a contradiction.
Claim 3 For each n ∈ N, if K ∩ A(n, 1) is infinite, then n ∈ K.
Now, let P = {{0}} ∪ {B(n, m) : n, m ∈ N} ∪ {{(n, m)} : n, m ∈ N}. Then P is a countable
collection of closed subsets of X. We show that P is a k-network of X.
Suppose that K ⊂ U with K compact and U open. By Claim 1, there exists k ∈ N such
that K ∩ N = {ni ∈ N : i ≤ k}. For each i ≤ k, ni ∈ K, so there exists B(ni , mi ) such
, : A Note on Weak Stratifiable Spaces
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that ni ∈ B(ni , mi ) ⊂ U . Set P1 = {B(ni , mi ) : i ≤ k}, then P1 is a finite subcollection of
P with K ∩ N ⊂ P1 ⊂ U . By Claim 2, the set I = {n ∈ N : K ∩ A(n, 1) = ∅} is finite.
Put I1 = {n ∈ I : K ∩ A(n, 1) is finite} and I2 = {n ∈ I : K ∩ A(n, 1) is infinite}. For each
n ∈ I1 , enumerate K ∩ A(n, 1) as {xni : i ≤ mn }, then for each i ≤ mn , xni ∈ {xni } ⊂ U and
{xni } ∈ P. Put P2 = {{xni } : i ≤ mn , n ∈ I1 }, then P2 is a finite subcollection of P with
K ∩ n∈I1 A(n, 1) ⊂ P2 ⊂ U . By Claim 3, for each n ∈ I2 , we have n ∈ K; thus there exists
B(n, kn ) such that n ∈ B(n, kn ) ⊂ U . Then K ∩A(n, 1)\B(n, kn ) is finite and we may enumerate
it as {yni : i ≤ jn }. Let P3 = {B(n, kn ) : n ∈ I2 } ∪ {{yni } : i ≤ jn , n ∈ I2 }, then P3 is a finite
subcollection of P with K ∩ n∈I2 A(n, 1) ⊂ P3 ⊂ U .
Now let P ∗ = P1 ∪ P2 ∪ P3 . If 0 ∈ K, then let P = P ∗ ∪ {{0}}; otherwise, let P = P ∗ .
In both case, P is a finite subcollection of P with K ⊂ P ⊂ U , which implies that P is a
k-network of X.
Proposition 1.3 Every space X that has a σ-cushioned pair sk-network is a weak stratifiable space.
Proof Let P = n∈N Pn be a pair sk-network for X, where Pn is cushioned and Pn ⊂ Pn+1
for each n ∈ N. For each x ∈ X and n ∈ N, put g(n, x) = X \ {P1 : (P1 , P2 ) ∈ Pn , x ∈
/ P2 }.
Since each Pn is cushioned, g is a g-function for X.
Now, suppose that x is a cluster point of yn and yn ∈ g(n, xn ) for all n ∈ N. If x is not a
cluster point of xn , then there is m ∈ N such that x ∈ X \ {xn : n ≥ m}. Since P = n∈N Pn
is a pair sk-network for X and Pn ⊂ Pn+1 for each n ∈ N, there exists k ≥ m and a finite
subcollection P of Pk such that x ∈ ( {P1 : (P1 , P2 ) ∈ P })◦ ⊂ {P2 : (P1 , P2 ) ∈ P } ⊂
X \ {xn : n ≥ m}. The fact that x is a cluster point of yn implies that there exists i ≥ k such
that yi ∈ ( {P1 : (P1 , P2 ) ∈ P })◦ . Hence, yi ∈ P1 for some (P1 , P2 ) ∈ P . Since xi ∈
/ P2 ,
/ g(k, xi ) and hence yi ∈
/ g(i, xi ), a contradiction.
we obtain that g(k, xi ) ∩ P1 = ∅. Thus, yi ∈
Therefore, x is a cluster point of xn and so X is a weak stratifiable space.
Consider the following conditions imposed on a g-function[1−4] .
(γ) If yn ∈ g(n, x) and xn ∈ g(n, yn ) for all n ∈ N, then x is a cluster point of xn ;
(Θ) If {x, xn } ⊂ g(n, yn ) for all n ∈ N and yn has a cluster point, then x is a cluster point
of xn ;
(ks) If yn ∈ g(n, xn ) for all n ∈ N and yn → x, then xn → x;
(developable) If {x, xn } ⊂ g(n, yn ) for all n ∈ N, then x is a cluster point of xn ;
A space that has a g-function satisfying condition (γ) (resp. (Θ)) is called a γ- (resp. (Θ-))
space while k-semi-stratifiable (resp. developable) spaces can be characterized by a g-function g
satisfying the condition (ks) (resp. (developable)).
It was shown in [10] that a k-semi-stratifiable space does not need to be a weak MCP space.
However, we have the following proposition.
Proposition 1.4 Every k-semi-stratifiable k-space X is a weak MCP space.
Proof Since X is a k-semi-stratifiable space, we get that every point of X is a Gδ -set.
Then, using the fact that X is a k-space, we obtain that X is a sequential space[8] .
Let g be a g-function satisfying condition (ks). Suppose that yn ∈ g(n, xn ) for all n ∈ N and
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p is a cluster point of yn . Since X is sequential, there is a subsequence ynk of yn such that
ynk converges to some point q. But ynk ∈ g(nk , xnk ) ⊂ g(k, xnk ) for all k ∈ N. Thus xnk → q
and q is a cluster point of xn . Therefore, X is a weak MCP space.
It is clear that both developable spaces and γ-spaces are Θ-spaces. In [9], weak MCP spaces
were called wcc-spaces and it was shown that a wcc-, γ-space (or a wcc-, developable space) is
metrizable. However, we have the following stronger result.
Theorem 1.5 Every weak MCP Θ-space is metrizable.
Proof It suffices to show that every weak MCP Θ-space is developable. Let h be a weak
MCP function and l be a Θ-function. For each x ∈ X and n ∈ N, put g(n, x) = h(n, x) ∩ l(n, x).
Suppose that {p, xn } ⊂ g(n, yn ) for all n ∈ N. Since p ∈ g(n, yn ) ⊂ h(n, yn ) and h is a weak
MCP function, yn has a cluster point. Now, {p, xn } ⊂ g(n, yn ) ⊂ l(n, yn ) and l is a Θ-function,
so p is a cluster point of xn . This implies that X is a developable space.
Acknowledgements The authors would like to thank the referee whose valuable comments have greatly improved the original manuscript.
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