Nonlinear Systems in Function Spaces and Applications in Biomedical Sciences, Control Theory, and EngineeringView this Special Issue
The -Property of Finite Unions of -Metrizable Spaces and Certain Function Spaces
The additivity of -property is studied on -metrizable spaces and certain function spaces. It is shown that a space of countable tightness is a -space provided that it is the union of finitely many -metrizable subspaces, or function spaces where each is Lindelöf .
1. Introduction and Definitions
The class of -spaces was introduced by van Douwen and Pfeffer in . It is well known that the extent coincides with the Lindelöf number in a -space, every countably compact -space is compact and every -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 ), Peng (see [7–9]), Fleissner and Stanley (see ), Soukup (see [11, 12]), Nyikos (see ), Alas et al. (see ), and so forth.
Among the topics for studying -spaces, the additivity of -property has been an important one since Arhagel'skii raised the question in  whether the union of two -subspaces is a -space. Recently, Soukup and Szeptycki constructed in  a non -space which is the union of two -subspaces. However, the answer is positive in some typical -classes (see [2, 4, 15–17]). Then it becomes an interesting work to find important -classes that preserve -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 -space if it is the union of finitely many -metrizable spaces, or function spaces where each is Lindelöf . 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 . To exhibit its importance, some more examples are shown in . In this paper, we use it creatively to deal with the finite unions of -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 spaces.
Firstly, we define for a set , and denote by the closure of in the whole space and by the closure of in the space . The symbol stands for the set of all positive natural numbers and for the real line equipped with the usual metric.
Definition 1 (see ). A relation from to is nearly good if implies for some .
Definition 2 (see ). Given a neighborhood assignment on , a subset of is -close if (equivalently, for every ).
Definition 3 (see ). A topological space is t-metrizable if there exists a metrizable topology on with and an assignment from to such that for every .
Definition 4 (see ). A cover of a topological space is thick if it satisfies the following condition:
One can assign to each so that for every .
Definition 5 (see ). A space has countable tightness if implies that for some countable subset of .
Definition 6 (see ). The topology of is called a point-open topology if the family is a subbase of the topology, where and is the topology of . The point-open topology is denoted by , and when , denoted by for short.
Definition 7 (see ). A Lindelöf -space is known as a -countably determined space, that is, there is a cover by compact sets and a countable collection such that, for any and , where is open in , then for some .
2. Finite Unions of -Metrizable Spaces and Function Spaces
In , Gruenhage introduced the concept of nearly good relations and build the following method to help discover -classes.
Proposition 8 (see ). Let be a neighborhood assignment on . Suppose there is a nearly good relation from to such that for any , is the countable union of -close sets. Then there is a closed and discrete set such that .
In , some interesting spaces are shown to be -spaces by constructing reasonable nearly good relations. Among them, -metrizable space is an important one. In this section, we use the method to discuss the relation between -property and the finite unions of -metrizable spaces. Note that it may be the first time to deal with finite unions of -spaces in this way.
Lemma 9. Suppose has countable tightness and , where each is -metrizable. Then is a -space.
Proof. For all , since is -metrizable, by [19, Theorem 3.4], let be the network of , where each is a thick partition of , and the assignment to each satisfies that, for every ,
For all and , we define in the following way.
For each , and , since and has countable tightness, there exists a , such that . Now let , and then we have that . Let , or , where . It is trivial that is a countable family. Furthermore, it satisfies the following condition.
Claim. For any , we have that .
To prove the claim, let . If , it follows from the thick property of that there is an and such that , and hence .
Or else, there exists , such that . Since has countable tightness, we can fix a such that . For every , we have by the construction of . Therefore, the following holds,
Let . Then there exists and such that . Assume that . For each , there exists an such that . Let . Then and . Consequently, .
Thus, we complete the proof of Claim and proceed to prove Lemma 9.
Now, for every , let . Then is also a countable family.
Let be an arbitrary neighborhood assignment on , and define a relation from to as follows,
To show that is nearly good, let and . There must exist such that . Without loss of generality, we assume that and . Since is a network of , there is an and such that . The following discussion help us know that is nearly good.(i)If , by the thick property of on , we have that . It follows that there exists a such that . Moreover, since is a partition of and , then , and hence . It witnesses that .(ii)If , by the foregoing claim, . There exists a such that . Then, by the definition of , there is a set and so that . Since and is a partition of . then this , and hence . Therefore, we have that .
By (i) and (ii), we know that is a nearly good relation.
For every , and , let . Then is -close. By the definition of the relation , it is easy to see that . Since is a countable family, is a countable union of -close sets.
By Proposition 8, there exists a closed and discrete subset of so that covers . It follows that is a -space.
Theorem 10. Suppose has countable tightness and , where each is -metrizable. Then is a -space.
Proof. By [23, Corollary 4.9], the result is true for . We prove inductively and assume that the result is true for many -metrizable subspaces.
Denote . It follows from Lemma 9 that is a -space. On the other hand, . For every , since countable tightness and -metrizability is hereditary, has countable tightness and it is the union of many -metrizable subspaces. By our assumption, each is a -space.
It is not difficult to check that, as the union of many open -subspaces, is a -space. Now we know that is a closed -subspace and is an open -subspace of . Then by [2, Proposition 1.2], is also a -space. Thus, the result is also true for many -metrizable subspaces.
Corollary 11. If a space from the following spaces is the union of finitely many -metrizable subspaces, then it is a -space.(a) First countable spaces;(b) Frechet-Urysohn spaces;(c) Sequential spaces.
Corollary 12. If a space of countable tightness is the union of finitely many spaces below, then it is a -space.(a) The free topological groups of a -metrizable Tychononoff spaces;(b) The -products of -metrizable spaces;(c) Hereditarily metaLindelöf descriptive spaces;(d) Spaces , where 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  the hereditary -property of , where is a compact Hausdorff space. It also follows from [19, Theorem 3.2] that the function space is -metrizable. Therefore, we obtain the result below.
Corollary 14. A space of countable tightness is a -space if it is the union of finitely many function spaces , where each is a compact Hausdorff space.
Moreover, in , Gruenhage generalized Buzyakova's result to the function space for the Lindelöf -space . 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 , where each is Lindelöf , then it is a -space.
Proof. Suppose that has countable tightness and , where each is Lindelöf . Based on the ideas shown in Lemma 9 and Theorem 10, it suffices to prove the result for and .
For with , and every , since , and has countable tightness, we fix such that . Then for every , let .
Since is Lindelöf , there is a cover by compact sets and a countable collection such that, for any and , where is open in , then for some .
Let be a countable base for the real line . For and , let . For , let be the set of all where and can be written in the form for some finite , where .
For every , let , and for every , let .
To show is a -space, let be a neighborhood assignment on . We define from to as follows,
For each and , let . Then is -close. Moreover, it is easy to check that is countable, and then is countable. It follows that is a countable union of -close sets.
To show is nearly good, let . Without loss of generality, we assume that and . Then for some countable set . Therefore, . As shown in the proof of [6, Proposition 2.7], there is a and such that . There must be an , such that . And hence . Thus, we have that , where .
By Proposition 8, there exists a closed and discrete subset of so that covers . This complete the proof of the -property of .
Conflict of Interests Statement
The authors declare that there is no conflict of interests regarding the publication of this article.
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.
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