Some results are obtained on finite unions of D-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.

1. Introduction

The concept of D-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 on D-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 studying D-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 be T1-spaces.

2. Definitions

For the purpose of convenience, we recall the following definitions.

Definition 1.

A neighborhood 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 is locally compact if each point in X has a compact neighborhood.

Definition 3.

A space is locally D 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 set A, define [A]<ω={H⊂A:|H|<ω}.

Definition 5 (see [<xref ref-type="bibr" rid="B7">7</xref>]).

A relation R 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 [<xref ref-type="bibr" rid="B6">6</xref>]).

Given a neighborhood assignment ϕ 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 [<xref ref-type="bibr" rid="B6">6</xref>]).

A family ℒ 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 [<xref ref-type="bibr" rid="B14">14</xref>]).

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 [<xref ref-type="bibr" rid="B14">14</xref>]).

A cover ℒ of a topological space X is thick if it satisfies the following condition.

One can assign ℒ(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 a T1-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 [<xref ref-type="bibr" rid="B15">15</xref>]).

A family 𝒫=⋃x∈X𝒫x of subsets of X is a weak base of X, if the following conditions holds.

For every x∈X, x∈∩𝒫x.

If U,V∈𝒫x, there exists W∈𝒫x such that W⊂U∩V.

A set U 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 by A¯ 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 Compact <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M122"><mml:mrow><mml:mi>D</mml:mi></mml:mrow></mml:math></inline-formula>-SpacesTheorem 12.

If a space X is the union of finitely many locally compact D-subspaces, then it is a D-space.

Proof.

Suppose that X=⋃i=1nXi, where each Xi is a locally compact D-space.

To prove that X 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 the D-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 set H=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 of X1, 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 set L=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 that D=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 case n>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 Theorem 12, we have the following consequence.

Corollary 13.

Suppose that X=⋃i=1nXi, where each Xi is a submetacompact locally compact subspace. Then X is a D-space.

Proof.

Since compact space is D 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 Theorem 12 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 that X=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 Discovering <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M258"><mml:mrow><mml:mi>D</mml:mi></mml:mrow></mml:math></inline-formula>-Classes

The following result presents us a method to discover D-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 [<xref ref-type="bibr" rid="B6">6</xref>]).

Let ϕ 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 a D-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 have D-property.

Proposition 16.

Every space with a point-countably expandable network is a D-space.

Proof.

Assume that X 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 that X 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 that R 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 each L∈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 [<xref ref-type="bibr" rid="B17">17</xref>]).

Every t-metrizable space is a D-space hereditarily.

Proof.

Suppose that X 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 that X 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 that R 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 that A¯⊂⋃{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.

Fix H∈[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 Proposition 15, 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 [<xref ref-type="bibr" rid="B18">18</xref>]).

Every space with a point-countable weak base is a D-space.

Proof.

Suppose that X 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 that X 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 that R 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 in 𝒫 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 by U 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.

Since x∈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.

Since 𝒫 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.

Acknowledgments

Xin Zhang is supported by the Natural Science Foundation of Shandong Province Grants ZR2010AQ001 and ZR2010AQ012, Hongfeng Guo is supported by the Natural Science Foundation of China Grants 11026108 and 11061004, and Yuming Xu is supported by the Natural Science Foundation of Shandong Province Grants ZR2010AM019, ZR2011AQ015, and 2012BSB01159.

van DouwenE. K.PfefferW. F.Some properties of the Sorgenfrey line and related spacesArhangel'skiiA. V.D-spaces and finite unionsArhangel'skiiA. V.D-spaces and covering propertiesArhangel'skiiA. V.BuzyakovaR. Z.Addition theorems and D-spacesBuzyakovaR. Z.Hereditary D-property of function spaces over compactaGruenhageG.A note on D-spacesPengL.-X.The D-property of some Lindelöf spaces and related conclusionsPengL.-X.A note on D-spaces and infinite unionsPengL.-X.On spaces which are D, linearly D and transitively DFleissnerW. G.StanleyA. M.D-spacesSoukupD. T.Constructing aD, non-D-spacesWorrellJ. M.Jr.WickeH. H.Characterizations of developable topological spacesJunnilaH.On submeta compact spacesDowA.JunnilaH.PelantJ.Coverings, networks and weak topologiesArkhangel'skiiA. V.Mappings and spacesEngelkingR.GuoH.JunnilaH.D-spaces and thick coversBurkeD. K.Weak-bases and D-spaces