61, 3 (2009), 241–246 September 2009
originalni nauqni rad research paper
CHARACTERIZATIONS OF δ-STRATIFIABLE SPACES Kedian Li
Abstract. In this paper, we give some characterizations ofδ-stratifiable spaces by means ofg-functions and semi-continuous functions. It is established that:
(1) A topological spaceXin which every point is a regularGδ-set isδ-stratifiable if and only if there exists ag-functiong:N×X→τsatisfies that ifF ∈RG(X) andy /∈F, then there is anm∈N such thaty /∈g(m, F);
(2) If there is an order preserving map ϕ : U SC(X) → LSC(X) such that for any h ∈ U SC(X),0≤ϕ(h)≤hand 0< ϕ(h)(x)< h(x) wheneverh(x)>0, thenX isδ-stratifiable space.
1. Introduction
It is one of the questions in general topology how to characterize the gen- eralized metric spaces [2, 5]. Recently, the problem of monotone insertions of generalized metric spaces has been studied [4]. Lane, Nyikos and Pan [7] proved that a topological spaceX is stratifiable if and only if there is an order-preserving map ψ:U L(X)→C(X) such that for any (g, h)∈U L(X),g ≤ψ(g, f)≤hand g(x)< ψ((g, h))(x)< h(x) whenever g(x)< h(x).
As a generalization of stratifiable spaces, Good and Haynes [3] defined δ- stratifiable spaces. Just as for stratifiablity, they discussed the products of compact metrizable spaces andδ-stratiable spaces. It is natural to pose the following ques- tion.
Question 1.1. How to characterizeδ-stratifiable spaces by the g-functions, or by the monotone insertion functions?
In this paper, we give some characterizations ofδ-stratifiable spaces by means ofg-functions and semi-continuous functions.
AMS Subject Classification: 54E20; 54C08; 26A15.
Keywords and phrases:δ-stratifiable spaces;g-functions; upper semi-continuous maps; lower semi-continuous maps.
The project is supported by the NNSF (Nos: 10571151, 10971185, 10971186) and FNSF (2006J0228) of China.
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All spaces in this paper are assumed to be T1. For a topological space X, τ denotes the topology on X, and τc ={X−O : O ∈ τ}. We refer the reader to [1, 8] for undefined terms.
A real-valued functionf defined on a spaceXislower (upper) semi-continuous if for eachx∈ X and each real number r with f(x)> r (f(x)< r), there exists a neighborhood U of x in X such that f(x0) > r (f(x0) < r) for every x0 ∈ U. We write LSC(X) (U SC(X)) for the set of all real-valued lower (upper) semi- continuous functions onX intoI= [0,1].
LetX be a space, ifA⊂X, we writeχAfor thecharacteristic functiononA.
Then χA ∈ U SC(X) if A is a closed subset of X, and χA ∈ LSC(X) if A is an open subset inX.
Ag-function on a topological space (X, τ) is a mappingg :N×X →τ such that x∈g(n, x) for each n∈N [6]. We define g(n, F) =S
{g(n, x) : x∈F} for eachF⊂X and eachn∈N.
We recall some basic concepts about theδ-stratifiable spaces.
Definition 1.2. [10] A subsetGof a topological spaceX is called a regular Gδ-set if G is an intersection of a sequence of closed sets whose interiors contain G, i.e. if G=T
n∈NFn =T
n∈NFn◦, whereFn is a closed set ofX. Equivalently, there exists a sequence {Un} of open sets such that G = T
n∈NUn =T
n∈NUn. The complement of a regularGδ-set is called a regular Fσ-set. Clearly, a set M is a regularFσ-set if and only if there exists a sequence{Fn}of closed sets such that M =S
n∈NFn =S
n∈NFn◦.
For a topological spaceX,RG(X) denotes the set of all regularGδ-sets ofX, andRF(X) ={X−G:G∈RG(X)} are the sets of all regularFσ-sets ofX.
Definition 1.3. [3] A topological spaceX isδ-stratifiable if and only if there is an operator U assigning to each n ∈N and D ∈RG(X), an open setU(n, D) containingDsuch that
(1) IfE, D∈RG(X) andE⊂D, then U(n, E)⊂U(n, D) for eachn∈N; (2) D=T
n∈NU(n, D).
We may assume that the operator U is also monotonic with respect to n, so thatU(n+ 1, D)⊂U(n, D) for eachn∈N and eachD∈RG(X).
The following lemma, included for convenience, is clearly just another way of stating the definition.
Lemma 1.4. A topological space X isδ-stratifiable if and only if there is an operator V :N×RF(X)→τc, such that
(1) F ⊃V(n, F)for eachF ∈RF(X)and alln∈N;
(2) If E, F ∈RF(X)andE⊂F, thenV(n, E)⊂V(n, F)for eachn∈N;
(3) F =S
n∈NV◦(n, F).
We may assume that V(n, F)⊂ V(n+ 1, F) for each n ∈ N and each F ∈ RF(X).
2. Main results and their proofs
First, we give a characterization ofδ-stratifiable spaces by theg-functions.
Lemma 2.1. If every point ofX is a regularGδ-set,X isδ-stratifiable if only and if there exists a g-function g :N×X →τ satisfying that if F ∈RG(X) and y /∈F, then there is an m∈N such that y /∈g(m, F).
Proof. Let X be δ-stratifiable and U an operator on X which satisfies con- ditions (1) and (2) in Definition 1.3. For each x ∈ X, let g(n, x) = U(n,{x}), then g : N ×X → τ is a g-function. Let F ∈ RG(X) and y /∈ F; we have y /∈F =T
n∈NU(n, F) by condition (2) of Definition 1.3. Thus there is anm∈N such thaty /∈U(m, F), and thereforey /∈g(m, F).
Conversely, suppose there exists ag-functiong:N×X →τ that satisfies the conditions given in the theorem. For eachD∈RG(X), let
U(n, D) =g(n, D) =[
{g(n, t) :t∈D}.
ThenU is an operation onX which satisfies the conditions (1) and (2) in Definition 1.3. In fact, it is clear for (1). For (2), if D ∈ RG(X) and D 6= T
n∈NU(n, D), there exists y ∈ T
n∈NU(n, D)−D. Since y /∈ D, there exists m ∈ N such that y /∈ U(m, D) by the condition of the theorem; this is a contradiction with y∈U(n, D) for eachn∈N.
Next, we characterizeδ-stratifiable spaces by semi-continuous functions.
Theorem 2.2. A space X is δ-stratifiable if and only if for any partially ordered set (H, <)and mapF :N×H→RG(X)such that
(1) F(n+ 1, h)⊂F(n, h)for allh∈H and alln∈N;
(2) for anyh1, h2∈H,if h1≤h2 thenF(n, h2)⊂F(n, h1),
there is a mapG:N×H→τ, such that (1) and (2) hold forG,F(n, h)⊂G(n, h) for allh∈H,n∈N andT
n∈NF(n, h) =T
n∈NG(n, h)for allh∈H.
Proof. Let X be a δ-stratifiable space and V an operator as in Lemma 1.4.
We show that the mapG:N×H→τ defined by
G(n, h) =X−V(n, X−F(n, h)),
satisfies the conditions of the theorem. By the properties of V and F, one can easily verify that the conditions (1) and (2) hold for G. SinceF(n, h)∈ RG(X) for each h∈Hand all n∈N, thenX−F(n, h)∈RF(X). By the condition (1) in Lemma 1.4,X−F(n, h)⊃V(n, X−F(n, h)) and soF(n, h)⊂G(n, h) for each h∈Hand alln∈N.
So we need only to show that T
n∈NF(n, h) =T
n∈NG(n, h) for all h∈ H.
If x /∈ T
n∈NF(n, h), then x /∈ F(m0, h) for some m0 ∈ N. Consequently, x ∈ V◦(n0, X −F(m0, h)) for some n0 ∈ N since X −F(m0, h) = S
n∈NV◦(n, X −
F(m0, h)). Letm = max{n0, m0}, then x ∈V◦(n0, X−F(m0, h))⊂ V(m, X − F(m0, h))⊂ V(m, X −F(m, h)), and x ∈ V◦(m, X−F(m, h)). But V(m, X − F(m, h))∩G(m, h) = ∅, hence x /∈ G(m, h), sox /∈ T
n∈NG(n, h), which proves the necessity.
Conversely, for eachD∈RF(X), consider the mapF :N×RF(X)→RG(X) defined byF(n, D) =X−D. One can easily verify thatFsatisfies the conditions (1) and (2) above. So there is a mapG:N×RF(X)→τ such that the conditions (1) and (2) hold forG. Moreover,F(n, D)⊂G(n, D) for alln∈Nand allD∈RF(X) and T
n∈NF(n, D) =T
n∈NG(n, D). Let V(n, D) =X −G(n, D), then the map V :N×RF(X)→τc satisfies the conditions in Lemma 1.4. In fact, it is clear that the condition (2) holds;V(n, D)⊂D becauseV(n, D) is a subset ofX−G(n, D), which is a closed subset ofX−F(n, D)=D, the condition (1) holds. We now show that the condition (3) holds. We only need to show thatD ⊂S
n∈NV◦(n, D). If x /∈ S
n∈NV◦(n, D), then x /∈ V◦(n, D) = X −X−V(n, D) = X−G(n, D) for alln∈N. This implies thatx∈T
n∈NG(n, D) =T
n∈NF(n, D) =X−D, hence x /∈D. SoX isδ-stratfiable.
Let (X, <) and (Y, <0) be a partially ordered sets. A mapψ:X →Y is said to beorder-preserving[1] ifψ(x)<0ψ(y) for every pairx, y∈X withx < y.
Theorem 2.3. Let X be a topological space. If there is an order preserving map ϕ: U SC(X)→LSC(X)such that for any h∈U SC(X),0 ≤ϕ(h)≤h and 0< ϕ(h)(x)< h(x) wheneverh(x)>0, thenX isδ-stratifiable.
Proof. Suppose that there is a mapϕ:U SC(X)→LSC(X) that satisfies the conditions of the theorem. For anyF ∈RF(X), F =S
n∈NWn =S
n∈NWn◦,Wn
is a closed subset ofX by Definition 1.2. Let hWn =χWn; thenhWn ∈U SC(X).
Let
hF(x) = P∞
n=1
1
2nhWn(x);
then hF ∈ U SC(X) by Theorem 2.4 in [11] and so ϕ(hF) ∈ LSC(X). For each n∈N, let
V(n, F) ={x∈X :ϕ(hF)(x)>1/2n} andV(n, F) ={x∈X:ϕ(hF)(x)>1/2n}.
Then the equality above defines a mapV :N×RF(X)→τc. We shall show that the mapV satisfies the conditions (1) through (3) in Lemma 1.4.
For eachn∈N, ifx∈V(n, F), then 1/2n < ϕ(hF)(x)≤hF(x). So hF(x) = Pn
k=1
1
2khWk(x) + P∞
k=n+1
1
2khWk(x) = P∞
n=1
1
2nhWn(x)>1/2n>0,
but P∞
k=n+1
1
2khWk(x)≤ P∞
k=n+1
1
2k = 1/2n.
Thus Pn
k=1 1
2khWk(x)>0. Hence there is k∈ {1,2,· · ·, n} such thatx∈ Wk ⊂ S
1≤k≤nWk, andV(n, F)⊂S
1≤k≤nWk. This implies that V(n, F)⊂ [
1≤k≤n
Wk⊂ [
1≤k≤n
Wk = [
1≤k≤n
Wk⊂F for eachn∈N and so
[
n∈N
V(n, F)⊂ [
n∈N
V(n, F)⊂F.
We show that reverse inclusion. Ifx /∈S
n∈NV(n, F), then x∈ \
n∈N
{t∈X:ϕ(hF)(t)≤1/2n}={t∈X:ϕ(hF)(t) = 0},
thusϕ(hF)(x) = 0. We havehF(x) = 0 by the property of the mapϕ. Hencex /∈F, and this implies thatF⊂S
n∈NV(n, F). SoF =S
n∈NV(n, F) =S
n∈NV(n, F).
If E, F ∈ RG(X), and E ⊂F, then hE ≤hF, andϕ(hE)≤ ϕ(hF). Hence V(n, E)⊂V(n, F) for all n∈N.
By Lemma 1.4,X isδ-strtifiable.
In the same manner as in Theorem 2.3, we can prove the following corollary.
Corollary 2.4. Let X be a topological space. If for each F ∈RF(X), there is anfF ∈LSC(X)that satisfies the following conditions:
(1) X−F =fF−1(0) and (2) fU ≤fV wheneverU ⊂V, thenX isδ-stratifiable.
For Theorem 2.3, we have a following question.
Question 2.5. Is there an order preserving map ϕ : U SC(X) → LSC(X) such that for any h∈U SC(X),0 ≤ϕ(h)≤hand 0 < ϕ(h)(x)< h(x) whenever h(x)>0, ifX isδ-stratifiable?
Acknowledgements. The author is very grateful to Professor Lin Shou for his help and like to thank the referee for the very constructive and valuable comments .
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(received 28.07.2008, in revised form 11.04.2009)
Department of Mathematics, Zhangzhou Normal University, Zhangzhou 363000, P. R. China E-mail:[email protected]
