# Finite Unions ofD-Spaces and Applications of  Nearly Good Relation

Authors: Xin Zhang; Hongfeng Guo; Yuming Xu
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## Abstract

Some results are obtained on finite unions ofD-spaces. It is proved
that if a space is the union of finitely many locally compact D-subspaces, then it
is a D-space. It follows that a space is a D-space if it is the union of finitely many
locally compact submetacompact subspaces. And a space is a D-space if it is the
union of a D-subspace with a locally compact D-subspace. This partially answers
one problem raised by Arhangel’skii. At last, some examples are given to exhibit
the applications of nearly good relation to discover D-classes.


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## Body

## 1. Introduction

The concept 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. Moreover, every countably compact D-space is compact and every D-space with the countable extent is Lindelöf. These facts make it helpful in studying covering properties. Some interesting work on D-spaces has been done by many topologists, especially by Arhangel’skii and Buzyakova (see [2–5]), Gruenhage (see [6]), Peng (see [7–9]), Fleissner and Stanley (see [10]), and Soukup (see [11]).Among the topics onD-spaces, the addition theorems occupy an important role. It has been an interesting subject, especially since Arhangel’skii raised the problem in [2, 3] whether the union of two D-subspaces is also D. In Section 3, we mainly consider the problem in locally compact spaces and give a partial answer by showing that a space is D provided that it is the union of a D-subspace with a locally compact D-subspace. Besides, we obtain that if a space is the union of finitely many locally compact D-subspaces, then it is a D-space. With its help, it is shown that a space is a D-space if it is the union of finitely many locally compact submetacompact subspaces.Another important task in studyingD-spaces is to discover typical D-classes. The method is a key to do this work, and hence some methods and related concepts emerged during the process. Among them, we believe that the concept of nearly good relation is an important one, which was introduced in [6] by  Gruenhage and helped to obtain that any space satisfying open G is D. Unfortunately, the concept did not attract much attention, and hence it is difficult to find other results based on the construction of nearly good relations. In fact, it can help us discover some D-classes easily. In Section 4, we exhibit this with some examples. Moreover, we hope that our work will interest others in studying D-spaces and related open problems.All spaces are assumed to beT1-spaces.


## 2. Definitions

For the purpose of convenience, we recall the following definitions.Definition 1.
Aneighborhood assignment is a mapping from a space to its topology. A space is called a D-space if, for every neighborhood assignment ϕ on X, there exists a closed discrete subset A of X such that the family ϕ(A)={ϕ(x):x∈A} covers X.Definition 2.
A space islocally compact if each point in X has a compact neighborhood.Definition 3.
A space islocallyD if each point in X has a neighborhood which is a D-space.In 1965, Worrell Jr. and Wicke in [12] introduced the class of θ-refinable spaces. Since this class generalizes paracompact spaces, metacompact spaces, and submetacompact spaces, Junnila suggested in [13] that this class be renamed submetacompact spaces; we use this name in the following.Definition 4.
A sequence〈ℒn〉 of covers of a space X is θ-sequence if, for every x∈X, there exists some n∈N such that the family ℒn is point finite at x. A space X is called submetacompact, if every open cover of X has a θ-sequence of open refinement.Note that, for a setA, define [A]<ω={H⊂A:|H|<ω}.Definition 5 (see [7]).
A relationR on a space X (resp., from X to [X]<ω) is nearly good if x∈A¯ implies xRy for some y∈A (resp., xRy~ for some y~∈[A]<ω).Definition 6 (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 7 (see [6]).
A familyℒ of subsets of a space X is point-countably expandable if there exists an open family {GL:L∈ℒ} such that L⊂GL for each L∈ℒ and {L∈ℒ:x∈GL} for each x∈X.Definition 8 (see [14]).
A topological space(X,𝒯) 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
(1)A¯𝒯⊂⋃H∈[A]<ω‍JH¯πforeveryA⊂X.Definition 9 (see [14]).
A coverℒ of a topological space X is thick if it satisfies the following condition.
One can assignℒ(H)∈ℒ<ω and L(H)=⋃ℒ(H) to each H∈[X]<ω so that
(2)A¯⊂⋃{L(H):H∈[A]<ω}foreveryA⊂X.Remark 10.
In aT1-space, the assignment “ℒ(H)∈ℒ<ω” in the previous condition can be weakened to “ℒ(H)∈ℒ≤ω” [14, Lemma 2.1]. Hence if (X,𝒯) is a T1-space, it is enough that the assignment H→JH is from [X]<ω to [X]≤ω in the definition of t-metrizable space.Definition 11 (see [15]).
A family𝒫=⋃x∈X𝒫x of subsets of X is a weak base of X, if the following conditions holds.(a)
For everyx∈X, x∈∩𝒫x.(b)
IfU,V∈𝒫x, there exists W∈𝒫x such that W⊂U∩V.(c)
A setU is open in X if and only if, for every x∈U, there exists P∈𝒫x such that P⊂U.In the following two sections, we denote byA¯ the closure of A in the whole space and by ClYA the closure of a set A in the space Y. Besides, denote by ℕ the set of all positive natural numbers.About other terminologies and notations that are omitted here, please refer to [16].


## 3. Finite Union of Locally CompactD-Spaces

Theorem 12.
If a spaceX is the union of finitely many locally compact D-subspaces, then it is a D-space.Proof.
Suppose thatX=⋃i=1n‍Xi, where each Xi is a locally compact D-space.
To prove thatX is a D-space, let ϕ be a neighborhood assignment on X. We prove inductively and suppose that X=X1∪X2 firstly.
Claim
1. X1 is open in X1¯.
Proof of Claim
1. Denote Y=X1¯, and take an x∈X1. Since X1 is locally compact, let U be an open neighborhood of x such that ClX1U is compact in X1. Choose an open subset V of Y with U=V∩X1. Then ClY(V∩X1)∩X1=ClYU∩X1=ClX1U. It follows that ClY(V∩X1)∩X1 is compact and hence closed in Y. Moreover, it contains V∩X1 and thus contains ClY(V∩X1). Then ClY(V∩X1)⊂X1, and hence (ClYX1)∩V⊂X1. It follows that (ClYX1)∩V=V is an open neighborhood of x in Y. Therefore, X1 is open in Y.
Claim
2. The set Z=X1¯∖X1 is closed in X.
Proof of Claim
2. Take an x∈X∖Z. If x∈X1, then V=(X∖X1¯)∪X1 is an open neighborhood in X such that V∩Z=∅. If x∈X2∖X1, then x∈X2∖X1¯, and assume on the contrary that x∈Z¯, which would follow that every neighborhood intersects Z and thus intersects X1, a contradiction with the fact x∉Z⊂X1¯.
As a closed subspace of theD-space X2, the space Z is a D-space. Then there exists a closed discrete subset D1 of Z, such that ∪ϕ(D1)⊃Z. Moreover, by Claim 2, Z is closed in X, so the set D1 is also closed and discrete in X.
The setH=X1∖∪ϕ(D1) is closed in X. Indeed, for any x∉H, we have that x∈∪ϕ(D1) or x∈X2∖∪ϕ(D1). If x∈∪ϕ(D1), then ∪ϕ(D1) is an open neighborhood of x missing H; if x∈X2∖∪ϕ(D1), then x∉X1, and hence x∉∪ϕ(D1)∪X1⊃Z∪X1=X1¯, so X∖X1¯ is an open neighborhood of x missing H.
As a closed subspace ofX1, H is also a D-space. There exists a closed discrete subset D2 in H, and thus in X, such that ∪ϕ(D2)⊃H.
Clearly, the setL=X∖∪ϕ(D1∪D2) is closed in X and contained in X2. Then there exists a closed discrete subset D3 in L, and thus in X, such that ∪ϕ(D3)⊃L.
It is trivial to check thatD=D1∪D2∪D3 is closed and discrete in X. Moreover, ϕ(D) is a cover of X since X=Z∪H∪L. Therefore, X is a D-space, and we complete the proof for the case n=2.
For the casen>2, assume inductively that ⋃i=1n-1‍Xi is a D-subspace. Since the subspace Xn is locally compact and open in its closure, with a similar construction as foregoing process, we can obtain a closed and discrete subset E of X such that ϕ(E) covers X. And thus X is a D-space.As a corollary of Theorem12, we have the following consequence.Corollary 13.
Suppose thatX=⋃i=1n‍Xi, where each Xi is a submetacompact locally compact subspace. Then X is a D-space.Proof.
Since compact space isD and Xi is locally compact for any 1≤i≤n, then every point of Xi has a neighborhood which is D-subset; that is, the space Xi is locally D. Moreover, because every locally submetacompact D-space is D [16, Theorem 5.10], each Xi is a D-space. By Theorem 12, as the union of finitely many locally compact D-spaces, the space X is a D-space.In fact, we see from the proof of Theorem12 that, when X=X1∪X2, the result can be obtained even only X1 or X2 is locally compact. So we have the following result, which is a partial answer to the problem whether a space is a D-space when the space is the union of two D-subspaces [3, Problem 1.1].Theorem 14.
Suppose thatX=X1∪X2, where X1 and X2 are all D-subspace and one of them is locally compact. Then X is a D-space.


## 4. Applications of Nearly Good Relation in DiscoveringD-Classes

The following result presents us a method to discoverD-spaces and we will show its use in this section with some examples. And we hope it will remind others with the use of nearly good relation in the study of D-spaces.Proposition 15 (see [6]).
Letϕ be a neighborhood assignment on X. Suppose that there is a nearly good relation R on X (or from X to [X]<ω) such that for any y∈X (or H∈[X]<ω), R-1(y)∖ϕ(y) (or R-1(H)∖∪ϕ(H)) is the countable union of ϕ-close sets. Then there is a closed discrete set D such that ∪ϕ(D)=X.It is well known that every space with countable base is aD-space. In this section, we mainly show that some general properties can also imply D.Firstly, the following result shows that many spaces with point-countable networks haveD-property.Proposition 16.
Every space with a point-countably expandable network is aD-space.Proof.
Assume thatX has a point-countably expandable network ℒ and the open family 𝒢={GL:L∈ℒ} satisfies that L⊂GL for each L∈ℒ and {L∈ℒ:x∈GL} is countable for each x∈X.
To show thatX is a D-space, let ϕ be an arbitrary neighborhood assignment on X. Define a relation on X as follows:
(3)xRy⟺∃L∈ℒ,suchthatx∈L⊂ϕ(x),y∈GL.
To show thatR is nearly good, let A⊂X, and let x∈A¯. Since ℒ is a network of x, there exists L∈ℒ such that x∈L. Then GL is an open neighborhood of x, and hence GL∩A≠∅. Then there exists y∈GL∩A. It follows that xRy and R are nearly good.
For eachL∈ℒ, let C(L)={x:x∈L⊂ϕ(x)}. Then for every x∈C(L), we have that ϕ(x)⊃C(L); that is, C(L) is a ϕ-close set. By the definition of the relation R, it is easy to check that R-1(y)=⋃y∈GL‍C(L) is a countable union of ϕ-close set. Hence by Proposition 15, there exists a closed discrete subset D of X such that ⋃ϕ(D)=X. And thus X is a D-space.In [14], a well-behaved class: t-metrizable spaces were introduced and then were proved in [17] to have D-property. Besides, as another good generalization of point-countable base, the point-countable weak base also implies D-property shown in [18]. However, the proofs of both results are very complicated. With the help of constructions of nearly good relations, we can prove them much easier.Proposition 17 (see [17]).
Everyt-metrizable space is a D-space hereditarily.Proof.
Suppose thatX is a t-metrizable space. Since every subspace of X is t-metrizable (see the remark following [14, Theorem 3.4]), we only need to show that X is a D-space.
By [14, Theorem 3.4], X has a network ℱ=⋃n∈ω‍ℱn, where each ℱn is a thick partition of X. Then for all n∈ω and H∈[X]<ω, let ℱn(H)∈ℱn<ω and Fn(H)=∪ℱn(H) be such that A¯⊂∪{Fn(H):H∈[A]<ω} for every A⊂X.
To show thatX is a D-space, let ϕ be an arbitrary neighborhood assignment on X. Define a relation from X to [X]<ω as follows:
(4)xRH⟺∃n∈ω,F∈ℱn(H),suchthatx∈F⊂ϕ(x).
To show thatR is nearly good, let A⊂X, and let x∈A¯. Since ℱ is a network of X, there exist m∈ω and Fx∈ℱm such that x∈Fx⊂ϕ(x).
We have thatA¯⊂⋃{Fm(H):H∈[A]<ω}, and it follows that there exists J∈[A]<ω such that x∈Fm(J)=⋃ℱm(J). Moreover, since ℱm is a partition of X and x∈Fx∈ℱm, then Fx∈ℱm(J). Therefore such m and Fx witness that xRH. We have shown that R is a nearly good relation.
FixH∈[X]<ω. For each n∈ω and F∈ℱn(H), let C(F)={x:x∈F⊂ϕ(x)}. For every x∈C(F), we have that ϕ(x)⊃C(F), and it follows that C(F) is a ϕ-close set. By the definition of the relation R, it is easy to check that R-1(H)=⋃n∈ω‍⋃F∈ℱn(H)‍C(F).
By Proposition15, there exists a closed discrete subset D of X such that ⋃ϕ(D)=X. We have shown that X is a D-space.Proposition 18 (see [18]).
Every space with a point-countable weak base is aD-space.Proof.
Suppose thatX has a weak base 𝒫={𝒫x:x∈X} such that {P∈𝒫:x∈P} is countable for every x∈X.
We call a finite family{Pxi:1≤i≤n} a chain of length n from x to y if, for every 1≤i≤n, xi+1∈Pxi for some Pxi∈𝒫xi where we denote x=x1 and y=xn+1.
To show thatX is a D-space, let ϕ be an arbitrary neighborhood assignment on X and define a relation on X in the following way:
(5)xRy⟺∃achain{Pxi:1≤i≤n}fromxtoy,wherex1=x.
To show thatR is nearly good, let A⊂X and x∈A¯. We construct a neighborhood of x as follows.
Step
1. Take a Px∈𝒫x.
Step
2. For every a∈Px taken in Step 1, take a Pa∈𝒫a.
Inductively, we take other sets in𝒫 in following steps. Assume that Step n-1 has been finished, and now we go to Step n.
Step n. For every P taken in Step n-1 and every b∈P, take a Pb∈𝒫b.
Denote byU the union of the set P taken in all steps. Then U is a open neighborhood of x. Indeed, for every z∈U, there must exist an n∈ℕ and P taken at Step n such that z∈P; then at Step n+1, one Pz∈𝒫z will be taken, and thus Pz⊂U.
Sincex∈A¯ and U is an open neighborhood of x, then there exists y∈A such that y∈U. It follows from the definition of R that xRy. Thereby, the relation R is nearly good.
Since𝒫 is point countable, then for each y∈X and n∈ℕ, the set Ln(y)={x∈X:xRy, and the length from x to y is n} is countable. It follows that R-1(y)=⋃n∈ℕ‍Ln(y) is countable, and hence it is the union of countable union of ϕ-close set.



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