# TheD-Property of Finite Unions of t-Metrizable Spaces and Certain Function Spaces

Authors: Xin Zhang; Hongfeng Guo
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

The additivity ofD-property is studied on t-metrizable spaces and
certain function spaces. It is shown that a space of countable tightness is a D-space
provided that it is the union of finitely many t-metrizable subspaces, or function spaces
Cp(Xi) where each Xi is Lindelöf Σ.


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## Body

## 1. Introduction and Definitions

The class ofD-spaces was introduced by van Douwen and Pfeffer in [1]. It is well known that the extent coincides with the Lindelöf number in a D-space, every countably compact D-space is compact and every D-space of countable extent is Lindelöf. These facts make it valuable as a covering property.A lot of work has been done these years by many topologists, especially by Arhangel'skii and Buzyakova (see [2–5]), Gruenhage (see [6]), Peng (see [7–9]), Fleissner and Stanley (see [10]), Soukup (see [11, 12]), Nyikos (see [13]), Alas et al. (see [14]), and so forth.Among the topics for studyingD-spaces, the additivity of D-property has been an important one since Arhagel'skii raised the question in [3] whether the union of two D-subspaces is a D-space. Recently, Soukup and Szeptycki constructed in [11] a T2 non D-space which is the union of two D-subspaces. However, the answer is positive in some typical D-classes (see [2, 4, 15–17]). Then it becomes an interesting work to find important D-classes that preserve D-property under finite unions. Motivated by this point, we try to discover some more general classes and obtain that a space of countable tightness is a D-space if it is the union of finitely many t-metrizable spaces, or function spaces Cp(Xi) where each Xi is Lindelöf Σ. It must be pointed out, in our work, we use the concept of nearly good relations, which was first introduced and well used by Gruenhage in [6]. To exhibit its importance, some more examples are shown in [18]. In this paper, we use it creatively to deal with the finite unions of D-spaces. We believe that more results will be obtained if we pay more attention to it.For convenience, we show some related definitions below. All spaces we consider in this paper are assumed to beT1 spaces.Firstly, we define[A]<ω={H⊂A:|H|<ω} for a set A, and denote by A¯ the closure of A in the whole space and by ClYA the closure of A in the space Y. The symbol ℕ stands for the set of all positive natural numbers and ℝ for the real line equipped with the usual metric.Definition 1 (see [6]).
A relationR from X to [X]<ω is nearly good if x∈A¯ implies xRH for some H∈[A]<ω.Definition 2 (see [6]).
Given a neighborhood assignmentϕ on X, a subset Z of X is ϕ-close if x,x′∈Z⇒x∈ϕ(x′) (equivalently, Z⊂ϕ(x) for every x∈Z).Definition 3 (see [19]).
A topological space(X,τ) is t-metrizable if there exists a metrizable topology π on X with τ⊂π and an assignment H↦JH from [X]<ω to [X]≤ω such that A¯τ⊂⋃H∈[A]<ω‍JH¯π for every A⊂X.Definition 4 (see [19]).
A coverℒ of a topological space X is thick if it satisfies the following condition:One can assignℒ(H)∈ℒ≤ω to each H∈[X]<ω so that A¯⊂⋃‍{⋃‍ℒ(H):H∈[A]<ω} for every A⊂X.Definition 5 (see [20]).
A spaceX has countable tightness if x∈A¯ implies that x∈C¯ for some countable subset C of A.Definition 6 (see [21]).
The topology ofC(X,L) is called a point-open topology if the family {[S,V]:S∈[X]<ω,V∈𝒯X} is a subbase of the topology, where [S,V]={f∈C(X,L):f(S)⊂V} and 𝒯X is the topology of X. The point-open topology is denoted by Cp(X,L), and when L=ℝ, denoted by Cp(X) for short.Definition 7 (see [22]).
ALindelöfΣ-space is known as a K-countably determined space, that is, there is a cover 𝒦 by compact sets and a countable collection ℱ such that, for any K∈𝒦 and K⊂U, where U is open in X, then K⊂F⊂U for some F∈ℱ.For other definitions and terminologies without showing here, please refer to [19–21].


## 2. Finite Unions oft-Metrizable Spaces and Function Spaces

In [6], Gruenhage introduced the concept of nearly good relations and build the following method to help discover D-classes.Proposition 8 (see [6]).
Letϕ be a neighborhood assignment on X. Suppose there is a nearly good relation from X to [X]<ω such that for any H∈[X]<ω, R-1(H)∖⋃‍ϕ(H) is the countable union of ϕ-close sets. Then there is a closed and discrete set D such that ⋃‍ϕ(D)=X.In [18], some interesting spaces are shown to be D-spaces by constructing reasonable nearly good relations. Among them, t-metrizable space is an important one. In this section, we use the method to discuss the relation between D-property and the finite unions of t-metrizable spaces. Note that it may be the first time to deal with finite unions of D-spaces in this way.Lemma 9.
SupposeX has countable tightness and X=⋃i=1m‍Xi, where each Xi is t-metrizable. Then Y=⋂i=1m‍Xi¯ is a D-space.Proof.
For alli∈{1,2,…,m}, since Xi is t-metrizable, by [19, Theorem 3.4], let ℒi=⋃n∈ℕ‍ℒi,n be the network of Xi, where each ℒi,n is a thick partition of Xi, and the assignment ℒi,n(H)∈[ℒi,n]≤ω to each H∈[Xi]<ω satisfies that, for every A⊂Xi,
(1)ClXiA⊂⋃‍{⋃‍ℒi,n(H):H∈[A]<ω}.
For alln∈ℕ and A∈[Y∩Xi]<ω, we define ℱi,n(A) in the following way.
For eachx∈A, and j≠i, since x∈Y∩Xj¯ and X has countable tightness, there exists a Bj(x)∈[Y∩Xj]≤ω, such that x∈Bj(x)¯. Now let Bj(A)=⋃‍{Bj(x):x∈A}, and then we have that A⊂Bj(A)¯. Let ℱi,n(A)={L∩Y:L∈ℒi,n(A), or L∈ℒj(n)(C), where C∈[Bj(A)]<ω,1≤j≤n,j≠i}. It is trivial that ℱi,n(A) is a countable family. Furthermore, it satisfies the following condition.
Claim. For any H⊂Xi∩Y, we have that ClYH⊂⋃‍{⋃‍ℱi,n(A):A∈[H]<ω}.
To prove the claim, letx∈ClYH. If x∈Xi, it follows from the thick property of ℒi,n that there is an A∈[H]<ω and L∈ℒi,n(A) such that x∈L, and hence x∈L∩Y∈ℱi,n(A).
Or else, there existsj≠i, such that x∈Xj. Since X has countable tightness, we can fix a K∈[H]≤ω such that x∈K¯. For every A∈[K]<ω, we have A⊂Bj(A)¯ by the construction of Bj(A). Therefore, the following holds,
(2)x∈K¯=⋃A∈[K]<ω‍A¯⊂⋃A∈[K]<ω‍Bj(A)¯¯=⋃A∈[K]<ω‍Bj(A)¯.
LetB=⋃A∈[K]<ω‍Bj(A). Then there exists C∈[B]<ω and L∈ℒj(n)(C) such that x∈L. Assume that C={yk:1≤k≤p}. For each yk, there exists an Ak∈[K]<ω such that yk∈Bj(Ak). Let A0=⋃1≤k≤p‍Ak. Then A0∈[K]<ω⊂[H]<ω and C∈[Bj(A0)]<ω. Consequently, x∈L∩Y∈ℱi,n(A0).
Thus, we complete the proof of Claim and proceed to prove Lemma9.
Now, for everyA∈[Y]<ω, let ℱn(A)=⋃i=1m‍ℱi,n(A∩Xi). Then ℱn(A) is also a countable family.
Letϕ be an arbitrary neighborhood assignment on Y, and define a relation R from Y to [Y]<ω as follows,
(3)xRH⟺∃n∈ℕ,L∩Y∈ℱn(H)(x∈L∩Y⊂ϕ(x)).
To show thatR is nearly good, let A⊂Y and x∈A¯. There must exist k,i∈{1,2,…,m} such that x∈Xk∩A∩Xi¯. Without loss of generality, we assume that i=1 and A⊂X1∩Y. Since ℒk=⋃n∈ℕ‍ℒk,n is a network of Xk, there is an n∈ℕ and Lx∈ℒk,n such that x∈Lx∩Y⊂ϕ(x). The following discussion help us know that R is nearly good.(i)
Ifk=1, by the thick property of ℒ1,n on X1, we have that ClX1A⊂⋃‍{⋃‍ℒ1,n(H):H∈[A]<ω}. It follows that there exists a J∈[A]<ω such that x∈⋃‍ℒ1,n(J). Moreover, since ℒ1,n is a partition of X1 and x∈Lx∈ℒ1,n, then Lx∈ℒ1,n(J), and hence Lx∩Y∈ℱ1,n(J). It witnesses that xRJ.(ii)
Ifk≠1, by the foregoing claim, ClYA⊂⋃‍{⋃‍ℱ1,n(H):H∈[A]<ω}. There exists a Q∈[A]<ω such that x∈⋃‍ℱ1,n(Q). Then, by the definition of ℱ1,n(Q), there is a set C∈[Bk(Q)]<ω and L∈ℒk,n(C) so that x∈L∩Y∈ℱ1,n(Q). Since ℒk,n(C)∈[ℒk,n]≤ω and ℒk,n is a partition of Xk. then this L=Lx, and hence x∈Lx∩Y∈ℱ1,n(Q). Therefore, we have that xRQ.
By (i) and (ii), we know thatR is a nearly good relation.
For everyH∈[Y]<ω, n∈ℕ and L∩Y∈ℱn(H), let (L∩Y)C={x∈R-1(H):x∈L∩Y⊂ϕ(x)}. Then (L∩Y)C is ϕ-close. By the definition of the relation R, it is easy to see that R-1(H)=⋃n∈ℕ‍⋃L∩Y∈ℱn(H)‍(L∩Y)C. Since ℱn(H) is a countable family, R-1(H) is a countable union of ϕ-close sets.
By Proposition8, there exists a closed and discrete subset D of Y so that ϕ(D) covers Y. It follows that Y is a D-space.Theorem 10.
SupposeX has countable tightness and X=⋃i=1m‍Xi, where each Xi is t-metrizable. Then X is a D-space.Proof.
By [23, Corollary 4.9], the result is true for m=1. We prove inductively and assume that the result is true for m-1 many t-metrizable subspaces.
DenoteY=⋂i=1m‍Xi¯. It follows from Lemma 9 that Y is a D-space. On the other hand, X∖Y=X∖⋂i=1m‍Xi¯=⋃i=1m‍(X∖Xi¯). For every i∈{1,2,…,m}, since countable tightness and t-metrizability is hereditary, X∖Xi¯ has countable tightness and it is the union of m-1 many t-metrizable subspaces. By our assumption, each X∖Xi¯ is a D-space.
It is not difficult to check that, as the union ofm many open D-subspaces, X∖Y is a D-space. Now we know that Y is a closed D-subspace and X∖Y is an open D-subspace of X. Then by [2, Proposition 1.2], X=(X∖Y)∪Y is also a D-space. Thus, the result is also true for m many t-metrizable subspaces.Since all first countable spaces, Frechet-Urysohn spaces and sequential spaces have countable tightness (see [20, Theorem  1.7.13]), we have the following consequence of Theorem 10.Corollary 11.
If a space from the following spaces is the union of finitely manyt-metrizable subspaces, then it is a D-space.(a)
First countable spaces;(b)
Frechet-Urysohn spaces;(c)
Sequential spaces.By [19, Theorem 3.2], some classes have t-metrizability, and then we can obtain the result below as another corollary of Theorem 10.Corollary 12.
If a space of countable tightness is the union of finitely many spaces below, then it is aD-space.(a)
The free topological groups of at-metrizable Tychononoff spaces;(b)
Theσ-products of t-metrizable spaces;(c)
Hereditarily metaLindelöf descriptive spaces;(d)
SpacesLw, where L is a metrizable locally convex vector space;(e)
Spaces with point-countably expandable networks.Remark 13.
The result concerning point-countably expandable networks answers [23, Problem 5.1] in a large part.Note that Buzyakova obtained in [5] the hereditary D-property of Cp(K), where K is a compact Hausdorff space. It also follows from [19, Theorem 3.2] that the function space Cp(K) is t-metrizable. Therefore, we obtain the result below.Corollary 14.
A space of countable tightness is aD-space if it is the union of finitely many function spaces Cp(Ki), where each Ki is a compact Hausdorff space.Moreover, in [6], Gruenhage generalized Buzyakova's result to the function space Cp(X) for the Lindelöf Σ-space X. Now using similar arguments as we have shown in the proofs of Lemma 9 and Theorem 10, we can generalize Corollary 14 to such function spaces. The idea of the construction of the nearly good relation in the proof below benefits from [6, Propositions 2.6 and 2.7].Theorem 15.
If a space of countable tightness is the union of finitely many function spacesCp(Xi), where each Xi is Lindelöf Σ, then it is a D-space.Proof.
Suppose thatY has countable tightness and Y=⋃i=1m‍Cp(Xi), where each Xi is Lindelöf Σ. Based on the ideas shown in Lemma 9 and Theorem 10, it suffices to prove the result for m=2 and Y=Cp(X1)¯∩Cp(X1)¯.
Fori,j∈{1,2} with i≠j, and every f∈Cp(Xi), since f∈Cp(Xj)¯, and Y has countable tightness, we fix Dj(f)∈[X]≤ω such that x∈Dj(f)¯. Then for every A⊂Cp(Xi), let Dj(A)=⋃‍{Dj(f):f∈A}.
SinceXi is Lindelöf Σ, there is a cover 𝒦i by compact sets and a countable collection ℱi such that, for any K∈𝒦i and K⊂U, where U is open in Xi, then K⊂F⊂U for some F∈ℱi.
Letℬ be a countable base for the real line ℝ. For S⊂Xi and B∈ℬ, let [S,B]i={f∈Cp(Xi):f(S)⊂B}. For A⊂C(Xi), let 𝒢i(A) be the set of all G=⋂k<n‍[Sk,Bk]i where Bk∈ℬ and Sk can be written in the form F∖⋃a∈A′‍a-1(Ba) for some finite A′⊂A, where F∈ℱi.
For everyH∈[Cp(Xi)]<ω, let 𝒢H=𝒢i(H)∪⋃‍{𝒢j(Q):Q∈[Dj(H)]<ω}, and for every H∈[Y]<ω, let 𝒢H=𝒢H∩Cp(X1)∪𝒢H∩Cp(X2).
To showY is a D-space, let ϕ be a neighborhood assignment on Y. We define R from Y to [Y]<ω as follows,
(4)fRH⟺∃G∈𝒢H(f∈G⊂ϕ(f)).
For eachH∈[Y]<ω and G∈𝒢H, let GC={f∈R-1(H):f∈G⊂ϕ(f)}. Then GC is ϕ-close. Moreover, it is easy to check that 𝒢H∩Cp(Xi) is countable, and then 𝒢H is countable. It follows that R-1(H)=⋃G∈𝒢H‍GC is a countable union of ϕ-close sets.
To showR is nearly good, let f∈A¯. Without loss of generality, we assume that f∈Cp(X1) and A⊂Cp(X2). Then f∈C¯ for some countable set C⊂A. Therefore, f∈C¯⊂D1(C)¯. As shown in the proof of [6, Proposition 2.7], there is a Q∈[D1(C)]<ω and G∈𝒢1(Q) such that f∈G⊂ϕ(f). There must be an H∈[C]<ω, such that Q∈[D1(H)]<ω. And hence G∈𝒢H. Thus, we have that fRH, where H∈[C]<ω⊂[A]<ω.
By Proposition8, there exists a closed and discrete subset E of Y so that ϕ(E) covers Y. This complete the proof of the D-property of Y.



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