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Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 808262, 4 pages
Finite Unions of -Spaces and Applications of Nearly Good Relation
1School of Mathematics and Quantitative Economics, Shandong University of Finance and Economics, Jinan, Shandong 250014, China
2School of Mathematics, Shandong University, Jinan, Shandong 250100, China
Received 14 January 2013; Accepted 7 April 2013
Academic Editor: Fuyi Xu
Copyright © 2013 Xin Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Some results are obtained on finite unions of -spaces. It is proved that if a space is the union of finitely many locally compact -subspaces, then it is a -space. It follows that a space is a -space if it is the union of finitely many locally compact submetacompact subspaces. And a space is a -space if it is the union of a -subspace with a locally compact -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 -classes.
The concept 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. Moreover, every countably compact -space is compact and every -space with the countable extent is Lindelöf. These facts make it helpful in studying covering properties. Some interesting work on -spaces has been done by many topologists, especially by Arhangel’skii and Buzyakova (see [2–5]), Gruenhage (see ), Peng (see [7–9]), Fleissner and Stanley (see ), and Soukup (see ).
Among the topics on -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 -subspaces is also . In Section 3, we mainly consider the problem in locally compact spaces and give a partial answer by showing that a space is provided that it is the union of a -subspace with a locally compact -subspace. Besides, we obtain that if a space is the union of finitely many locally compact -subspaces, then it is a -space. With its help, it is shown that a space is a -space if it is the union of finitely many locally compact submetacompact subspaces.
Another important task in studying -spaces is to discover typical -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  by Gruenhage and helped to obtain that any space satisfying open is . 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 -classes easily. In Section 4, we exhibit this with some examples. Moreover, we hope that our work will interest others in studying -spaces and related open problems.
All spaces are assumed to be -spaces.
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 -space if, for every neighborhood assignment on , there exists a closed discrete subset of such that the family covers .
Definition 2. A space is locally compact if each point in has a compact neighborhood.
Definition 3. A space is locally if each point in has a neighborhood which is a -space.
In 1965, Worrell Jr. and Wicke in  introduced the class of -refinable spaces. Since this class generalizes paracompact spaces, metacompact spaces, and submetacompact spaces, Junnila suggested in  that this class be renamed submetacompact spaces; we use this name in the following.
Definition 4. A sequence of covers of a space is -sequence if, for every , there exists some such that the family is point finite at . A space is called submetacompact, if every open cover of has a -sequence of open refinement.
Note that, for a set , define .
Definition 5 (see ). A relation on a space (resp., from to ) is nearly good if implies for some (resp., for some ).
Definition 6 (see ). Given a neighborhood assignment on , a subset of is -close if (equivalently, for every ).
Definition 7 (see ). A family of subsets of a space is point-countably expandable if there exists an open family such that for each and for each .
Definition 8 (see ). A topological space is -metrizable if there exists a metrizable topology on with and an assignment from to such that
Definition 9 (see ). A cover of a topological space is thick if it satisfies the following condition.
One can assign and to each so that
Remark 10. In a -space, the assignment “” in the previous condition can be weakened to “” [14, Lemma 2.1]. Hence if is a -space, it is enough that the assignment is from to in the definition of -metrizable space.
Definition 11 (see ). A family of subsets of is a weak base of , if the following conditions holds.(a)For every , .(b)If , there exists such that .(c)A set is open in if and only if, for every , there exists such that .
In the following two sections, we denote by the closure of in the whole space and by the closure of a set in the space . Besides, denote by the set of all positive natural numbers.
About other terminologies and notations that are omitted here, please refer to .
3. Finite Union of Locally Compact -Spaces
Theorem 12. If a space is the union of finitely many locally compact -subspaces, then it is a -space.
Proof. Suppose that , where each is a locally compact -space.
To prove that is a -space, let be a neighborhood assignment on . We prove inductively and suppose that firstly.
Claim 1. is open in .
Proof of Claim 1. Denote , and take an . Since is locally compact, let be an open neighborhood of such that is compact in . Choose an open subset of with . Then . It follows that is compact and hence closed in . Moreover, it contains and thus contains . Then , and hence . It follows that is an open neighborhood of in . Therefore, is open in .
Claim 2. The set is closed in .
Proof of Claim 2. Take an . If , then is an open neighborhood in such that . If , then , and assume on the contrary that , which would follow that every neighborhood intersects and thus intersects , a contradiction with the fact .
As a closed subspace of the -space , the space is a -space. Then there exists a closed discrete subset of , such that . Moreover, by Claim 2, is closed in , so the set is also closed and discrete in .
The set is closed in . Indeed, for any , we have that or . If , then is an open neighborhood of missing ; if , then , and hence , so is an open neighborhood of missing .
As a closed subspace of , is also a -space. There exists a closed discrete subset in , and thus in , such that .
Clearly, the set is closed in and contained in . Then there exists a closed discrete subset in , and thus in , such that .
It is trivial to check that is closed and discrete in . Moreover, is a cover of since . Therefore, is a -space, and we complete the proof for the case .
For the case , assume inductively that is a -subspace. Since the subspace is locally compact and open in its closure, with a similar construction as foregoing process, we can obtain a closed and discrete subset of such that covers . And thus is a -space.
As a corollary of Theorem 12, we have the following consequence.
Corollary 13. Suppose that , where each is a submetacompact locally compact subspace. Then is a -space.
Proof. Since compact space is and is locally compact for any , then every point of has a neighborhood which is -subset; that is, the space is locally . Moreover, because every locally submetacompact -space is [16, Theorem 5.10], each is a -space. By Theorem 12, as the union of finitely many locally compact -spaces, the space is a -space.
In fact, we see from the proof of Theorem 12 that, when , the result can be obtained even only or is locally compact. So we have the following result, which is a partial answer to the problem whether a space is a -space when the space is the union of two -subspaces [3, Problem 1.1].
Theorem 14. Suppose that , where and are all -subspace and one of them is locally compact. Then is a -space.
4. Applications of Nearly Good Relation in Discovering -Classes
The following result presents us a method to discover -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 -spaces.
Proposition 15 (see ). Let be a neighborhood assignment on . Suppose that there is a nearly good relation on (or from to ) such that for any (or ), (or ) is the countable union of -close sets. Then there is a closed discrete set such that .
It is well known that every space with countable base is a -space. In this section, we mainly show that some general properties can also imply .
Firstly, the following result shows that many spaces with point-countable networks have -property.
Proposition 16. Every space with a point-countably expandable network is a -space.
Proof. Assume that has a point-countably expandable network and the open family satisfies that for each and is countable for each .
To show that is a -space, let be an arbitrary neighborhood assignment on . Define a relation on as follows:
To show that is nearly good, let , and let . Since is a network of , there exists such that . Then is an open neighborhood of , and hence . Then there exists . It follows that and are nearly good.
For each , let . Then for every , we have that ; that is, is a -close set. By the definition of the relation , it is easy to check that is a countable union of -close set. Hence by Proposition 15, there exists a closed discrete subset of such that . And thus is a -space.
In , a well-behaved class: -metrizable spaces were introduced and then were proved in  to have -property. Besides, as another good generalization of point-countable base, the point-countable weak base also implies -property shown in . 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 ). Every -metrizable space is a -space hereditarily.
Proof. Suppose that is a -metrizable space. Since every subspace of is -metrizable (see the remark following [14, Theorem 3.4]), we only need to show that is a -space.
By [14, Theorem 3.4], has a network , where each is a thick partition of . Then for all and , let and be such that for every .
To show that is a -space, let be an arbitrary neighborhood assignment on . Define a relation from to as follows:
To show that is nearly good, let , and let . Since is a network of , there exist and such that .
We have that , and it follows that there exists such that . Moreover, since is a partition of and , then . Therefore such and witness that . We have shown that is a nearly good relation.
Fix . For each and , let . For every , we have that , and it follows that is a -close set. By the definition of the relation , it is easy to check that .
By Proposition 15, there exists a closed discrete subset of such that . We have shown that is a -space.
Proposition 18 (see ). Every space with a point-countable weak base is a -space.
Proof. Suppose that has a weak base such that is countable for every .
We call a finite family a chain of length from to if, for every , for some where we denote and .
To show that is a -space, let be an arbitrary neighborhood assignment on and define a relation on in the following way:
To show that is nearly good, let and . We construct a neighborhood of as follows.
Step 1. Take a .
Step 2. For every taken in Step 1, take a .
Inductively, we take other sets in in following steps. Assume that Step has been finished, and now we go to Step .
Step n. For every taken in Step and every , take a .
Denote by the union of the set taken in all steps. Then is a open neighborhood of . Indeed, for every , there must exist an and taken at Step such that ; then at Step , one will be taken, and thus .
Since and is an open neighborhood of , then there exists such that . It follows from the definition of that . Thereby, the relation is nearly good.
Since is point countable, then for each and , the set , and the length from to is is countable. It follows that is countable, and hence it is the union of countable union of -close set.
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.
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