The 𝐷-Property of Finite Unions of 𝑡-Metrizable Spaces and Certain Function Spaces

Abstract

The additivity of 𝐷-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 𝐶_𝑝(𝑋_𝑖) where each 𝑋_𝑖 is Lindelöf Σ.

1. Introduction and Definitions

The class 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, 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 studying 𝐷-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 𝐶_𝑝(𝑋_𝑖) where each 𝑋_𝑖 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 be 𝑇₁ 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 Cl_𝑌𝐴 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 relation 𝑅 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 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 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↦𝐽_𝐻 from [𝑋]^<𝜔 to [𝑋]^≤𝜔 such that 𝐴^𝜏⊂⋃_{𝐻∈[𝐴]}^<𝜔𝐽_𝐻^𝜋 for every 𝐴⊂𝑋.

Definition 4 (see [19]). A cover ℒ of a topological space 𝑋 is thick if it satisfies the following condition:

One can assign ℒ(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 space 𝑋 has countable tightness if 𝑥∈𝐴 implies that 𝑥∈𝐶 for some countable subset 𝐶 of 𝐴.

Definition 6 (see [21]). The topology of 𝐶(𝑋,𝐿) is called a point-open topology if the family {[𝑆,𝑉]∶𝑆∈[𝑋]^<𝜔,𝑉∈𝒯_𝑋} is a subbase of the topology, where [𝑆,𝑉]={𝑓∈𝐶(𝑋,𝐿)∶𝑓(𝑆)⊂𝑉} and 𝒯_𝑋 is the topology of 𝑋. The point-open topology is denoted by 𝐶_𝑝(𝑋,𝐿), and when 𝐿=ℝ, denoted by 𝐶_𝑝(𝑋) for short.

Definition 7 (see [22]). A Lindelöf Σ-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 of 𝑡-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]). Let 𝜙 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. Suppose 𝑋 has countable tightness and 𝑋=⋃^𝑚_𝑖=1𝑋_𝑖, where each 𝑋_𝑖 is 𝑡-metrizable. Then 𝑌=⋂^𝑚_𝑖=1𝑋_𝑖 is a 𝐷-space.

Proof. For all 𝑖∈{1,2,…,𝑚}, since 𝑋_𝑖 is 𝑡-metrizable, by [19, Theorem 3.4], let ℒ_𝑖=⋃_𝑛∈ℕℒ_𝑖,𝑛 be the network of 𝑋_𝑖, where each ℒ_𝑖,𝑛 is a thick partition of 𝑋_𝑖, and the assignment ℒ_𝑖,𝑛(𝐻)∈[ℒ_𝑖,𝑛]^≤𝜔 to each 𝐻∈[𝑋_𝑖]^<𝜔 satisfies that, for every 𝐴⊂𝑋_𝑖, Cl_𝑋𝑖𝐴⊂ℒ_𝑖,𝑛(𝐻)∶𝐻∈[𝐴]^<𝜔. (1)
For all 𝑛∈ℕ and 𝐴∈[𝑌∩𝑋_𝑖]^<𝜔, we define ℱ_𝑖,𝑛(𝐴) in the following way.
For each 𝑥∈𝐴, and 𝑗≠𝑖, since 𝑥∈𝑌∩𝑋_𝑗 and 𝑋 has countable tightness, there exists a 𝐵_𝑗(𝑥)∈[𝑌∩𝑋_𝑗]^≤𝜔, such that 𝑥∈𝐵_𝑗(𝑥). Now let 𝐵_𝑗(𝐴)=⋃{𝐵_𝑗(𝑥)∶𝑥∈𝐴}, and then we have that 𝐴⊂𝐵_𝑗(𝐴). Let ℱ_𝑖,𝑛(𝐴)={𝐿∩𝑌∶𝐿∈ℒ_𝑖,𝑛(𝐴), or 𝐿∈ℒ_𝑗(𝑛)(𝐶), where 𝐶∈[𝐵_𝑗(𝐴)]^<𝜔,1≤𝑗≤𝑛,𝑗≠𝑖}. It is trivial that ℱ_𝑖,𝑛(𝐴) is a countable family. Furthermore, it satisfies the following condition.
Claim. For any 𝐻⊂𝑋_𝑖∩𝑌, we have that Cl_𝑌𝐻⊂⋃{⋃ℱ_𝑖,𝑛(𝐴)∶𝐴∈[𝐻]^<𝜔}.
To prove the claim, let 𝑥∈Cl_𝑌𝐻. If 𝑥∈𝑋_𝑖, it follows from the thick property of ℒ_𝑖,𝑛 that there is an 𝐴∈[𝐻]^<𝜔 and 𝐿∈ℒ_𝑖,𝑛(𝐴) such that 𝑥∈𝐿, and hence 𝑥∈𝐿∩𝑌∈ℱ_𝑖,𝑛(𝐴).
Or else, there exists 𝑗≠𝑖, such that 𝑥∈𝑋_𝑗. Since 𝑋 has countable tightness, we can fix a 𝐾∈[𝐻]^≤𝜔 such that 𝑥∈𝐾. For every 𝐴∈[𝐾]^<𝜔, we have 𝐴⊂𝐵_𝑗(𝐴) by the construction of 𝐵_𝑗(𝐴). Therefore, the following holds, 𝑥∈𝐾=_{[𝐾]𝐴∈<𝜔}𝐴⊂_{[𝐾]𝐴∈<𝜔}𝐵_𝑗(𝐴)=_{[𝐾]𝐴∈<𝜔}𝐵_𝑗(𝐴). (2)
Let 𝐵=⋃_{𝐴∈[𝐾]}^<𝜔𝐵_𝑗(𝐴). Then there exists 𝐶∈[𝐵]^<𝜔 and 𝐿∈ℒ_𝑗(𝑛)(𝐶) such that 𝑥∈𝐿. Assume that 𝐶={𝑦_𝑘∶1≤𝑘≤𝑝}. For each 𝑦_𝑘, there exists an 𝐴_𝑘∈[𝐾]^<𝜔 such that 𝑦_𝑘∈𝐵_𝑗(𝐴_𝑘). Let 𝐴₀=⋃_{1≤𝑘≤𝑝}𝐴_𝑘. Then 𝐴₀∈[𝐾]^<𝜔⊂[𝐻]^<𝜔 and 𝐶∈[𝐵_𝑗(𝐴₀)]^<𝜔. Consequently, 𝑥∈𝐿∩𝑌∈ℱ_𝑖,𝑛(𝐴₀).
Thus, we complete the proof of Claim and proceed to prove Lemma 9.
Now, for every 𝐴∈[𝑌]^<𝜔, let ℱ_𝑛(𝐴)=⋃^𝑚_𝑖=1ℱ_𝑖,𝑛(𝐴∩𝑋_𝑖). Then ℱ_𝑛(𝐴) is also a countable family.
Let 𝜙 be an arbitrary neighborhood assignment on 𝑌, and define a relation 𝑅 from 𝑌 to [𝑌]^<𝜔 as follows, 𝑥𝑅𝐻⟺∃𝑛∈ℕ,𝐿∩𝑌∈ℱ_𝑛(𝐻)𝑥∈𝐿∩𝑌⊂𝜙(𝑥). (3)
To show that 𝑅 is nearly good, let 𝐴⊂𝑌 and 𝑥∈𝐴. There must exist 𝑘,𝑖∈{1,2,…,𝑚} such that 𝑥∈𝑋_𝑘∩𝐴∩𝑋_𝑖. Without loss of generality, we assume that 𝑖=1 and 𝐴⊂𝑋₁∩𝑌. Since ℒ_𝑘=⋃_𝑛∈ℕℒ_𝑘,𝑛 is a network of 𝑋_𝑘, there is an 𝑛∈ℕ and 𝐿_𝑥∈ℒ_𝑘,𝑛 such that 𝑥∈𝐿_𝑥∩𝑌⊂𝜙(𝑥). The following discussion help us know that 𝑅 is nearly good.(i)If 𝑘=1, by the thick property of ℒ_1,𝑛 on 𝑋₁, we have that Cl_𝑋1𝐴⊂⋃{⋃ℒ_1,𝑛(𝐻)∶𝐻∈[𝐴]^<𝜔}. It follows that there exists a 𝐽∈[𝐴]^<𝜔 such that 𝑥∈⋃ℒ_1,𝑛(𝐽). Moreover, since ℒ_1,𝑛 is a partition of 𝑋₁ and 𝑥∈𝐿_𝑥∈ℒ_1,𝑛, then 𝐿_𝑥∈ℒ_1,𝑛(𝐽), and hence 𝐿_𝑥∩𝑌∈ℱ_1,𝑛(𝐽). It witnesses that 𝑥𝑅𝐽.(ii)If 𝑘≠1, by the foregoing claim, Cl_𝑌𝐴⊂⋃{⋃ℱ_1,𝑛(𝐻)∶𝐻∈[𝐴]^<𝜔}. There exists a 𝑄∈[𝐴]^<𝜔 such that 𝑥∈⋃ℱ_1,𝑛(𝑄). Then, by the definition of ℱ_1,𝑛(𝑄), there is a set 𝐶∈[𝐵_𝑘(𝑄)]^<𝜔 and 𝐿∈ℒ_𝑘,𝑛(𝐶) so that 𝑥∈𝐿∩𝑌∈ℱ_1,𝑛(𝑄). Since ℒ_𝑘,𝑛(𝐶)∈[ℒ_𝑘,𝑛]^≤𝜔 and ℒ_𝑘,𝑛 is a partition of 𝑋_𝑘. then this 𝐿=𝐿_𝑥, and hence 𝑥∈𝐿_𝑥∩𝑌∈ℱ_1,𝑛(𝑄). Therefore, we have that 𝑥𝑅𝑄.
By (i) and (ii), we know that 𝑅 is a nearly good relation.
For every 𝐻∈[𝑌]^<𝜔, 𝑛∈ℕ and 𝐿∩𝑌∈ℱ_𝑛(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 ℱ_𝑛(𝐻) is a countable family, 𝑅⁻¹(𝐻) is a countable union of 𝜙-close sets.
By Proposition 8, 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. Suppose 𝑋 has countable tightness and 𝑋=⋃^𝑚_𝑖=1𝑋_𝑖, where each 𝑋_𝑖 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.
Denote 𝑌=⋂^𝑚_𝑖=1𝑋_𝑖. It follows from Lemma 9 that 𝑌 is a 𝐷-space. On the other hand, 𝑋⧵𝑌=𝑋⧵⋂^𝑚_𝑖=1𝑋_𝑖=⋃^𝑚_𝑖=1(𝑋⧵𝑋_𝑖). For every 𝑖∈{1,2,…,𝑚}, since countable tightness and 𝑡-metrizability is hereditary, 𝑋⧵𝑋_𝑖 has countable tightness and it is the union of 𝑚−1 many 𝑡-metrizable subspaces. By our assumption, each 𝑋⧵𝑋_𝑖 is a 𝐷-space.
It is not difficult to check that, as the union of 𝑚 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 many 𝑡-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 a 𝐷-space.(a) The free topological groups of a 𝑡-metrizable Tychononoff spaces;(b) The 𝜎-products of 𝑡-metrizable spaces;(c) Hereditarily metaLindelöf descriptive spaces;(d) Spaces 𝐿_𝑤, 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 𝐶_𝑝(𝐾), where 𝐾 is a compact Hausdorff space. It also follows from [19, Theorem 3.2] that the function space 𝐶_𝑝(𝐾) is 𝑡-metrizable. Therefore, we obtain the result below.

Corollary 14. A space of countable tightness is a 𝐷-space if it is the union of finitely many function spaces 𝐶_𝑝(𝐾_𝑖), where each 𝐾_𝑖 is a compact Hausdorff space.

Moreover, in [6], Gruenhage generalized Buzyakova's result to the function space 𝐶_𝑝(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 spaces 𝐶_𝑝(𝑋_𝑖), where each 𝑋_𝑖 is Lindelöf Σ, then it is a 𝐷-space.

Proof. Suppose that 𝑌 has countable tightness and 𝑌=⋃^𝑚_𝑖=1𝐶_𝑝(𝑋_𝑖), where each 𝑋_𝑖 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=𝐶_𝑝(𝑋₁)∩𝐶_𝑝(𝑋₁).
For 𝑖,𝑗∈{1,2} with 𝑖≠𝑗, and every 𝑓∈𝐶_𝑝(𝑋_𝑖), since 𝑓∈𝐶_𝑝(𝑋_𝑗), and 𝑌 has countable tightness, we fix 𝐷_𝑗(𝑓)∈[𝑋]^≤𝜔 such that 𝑥∈𝐷_𝑗(𝑓). Then for every 𝐴⊂𝐶_𝑝(𝑋_𝑖), let 𝐷_𝑗(𝐴)=⋃{𝐷_𝑗(𝑓)∶𝑓∈𝐴}.
Since 𝑋_𝑖 is Lindelöf Σ, there is a cover 𝒦_𝑖 by compact sets and a countable collection ℱ_𝑖 such that, for any 𝐾∈𝒦_𝑖 and 𝐾⊂𝑈, where 𝑈 is open in 𝑋_𝑖, then 𝐾⊂𝐹⊂𝑈 for some 𝐹∈ℱ_𝑖.
Let ℬ be a countable base for the real line ℝ. For 𝑆⊂𝑋_𝑖 and 𝐵∈ℬ, let [𝑆,𝐵]_𝑖={𝑓∈𝐶_𝑝(𝑋_𝑖)∶𝑓(𝑆)⊂𝐵}. For 𝐴⊂𝐶(𝑋_𝑖), let 𝒢_𝑖(𝐴) be the set of all 𝐺=⋂_𝑘<𝑛[𝑆_𝑘,𝐵_𝑘]_𝑖 where 𝐵_𝑘∈ℬ and 𝑆_𝑘 can be written in the form 𝐹⧵⋃_{𝑎∈𝐴′}𝑎⁻¹(𝐵_𝑎) for some finite 𝐴^′⊂𝐴, where 𝐹∈ℱ_𝑖.
For every 𝐻∈[𝐶_𝑝(𝑋_𝑖)]^<𝜔, let 𝒢_𝐻=𝒢_𝑖(𝐻)∪⋃{𝒢_𝑗(𝑄)∶𝑄∈[𝐷_𝑗(𝐻)]^<𝜔}, and for every 𝐻∈[𝑌]^<𝜔, let 𝒢_𝐻=𝒢_{𝐻∩𝐶𝑝(𝑋1)}∪𝒢_{𝐻∩𝐶𝑝(𝑋2)}.
To show 𝑌 is a 𝐷-space, let 𝜙 be a neighborhood assignment on 𝑌. We define 𝑅 from 𝑌 to [𝑌]^<𝜔 as follows, 𝑓𝑅𝐻⟺∃𝐺∈𝒢_𝐻𝑓∈𝐺⊂𝜙𝑓. (4)
For each 𝐻∈[𝑌]^<𝜔 and 𝐺∈𝒢_𝐻, let 𝐺_𝐶={𝑓∈𝑅⁻¹(𝐻)∶𝑓∈𝐺⊂𝜙(𝑓)}. Then 𝐺_𝐶 is 𝜙-close. Moreover, it is easy to check that 𝒢_{𝐻∩𝐶𝑝(𝑋𝑖)} is countable, and then 𝒢_𝐻 is countable. It follows that 𝑅⁻¹(𝐻)=⋃_{𝐺∈𝒢𝐻}𝐺_𝐶 is a countable union of 𝜙-close sets.
To show 𝑅 is nearly good, let 𝑓∈𝐴. Without loss of generality, we assume that 𝑓∈𝐶_𝑝(𝑋₁) and 𝐴⊂𝐶_𝑝(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∈𝒢_1(𝑄) such that 𝑓∈𝐺⊂𝜙(𝑓). There must be an 𝐻∈[𝐶]^<𝜔, such that 𝑄∈[𝐷₁(𝐻)]^<𝜔. And hence 𝐺∈𝒢_𝐻. Thus, we have that 𝑓𝑅𝐻, where 𝐻∈[𝐶]^<𝜔⊂[𝐴]^<𝜔.
By Proposition 8, 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.

Conflict of Interests Statement

The authors declare that there is no conflict of interests regarding the publication of this article.

Acknowledgments

Supported by Natural Science Foundation of Shandong Province grant ZR2010AQ001. Supported by Natural Science Foundation of China grant 11026108 and 11061004, and by Natural Science Foundation of Shandong Province grant ZR2010AQ012.