Semi-<span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.06580067pt" id="M1" height="11.1379pt" version="1.1" viewBox="-0.0657574 -11.0721 6.00502 11.1379" width="6.00502pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g121-71" d="M303 0V28C221 34 213 39 213 125V525C213 610 221 616 303 622V650H38V622C120 616 128 610 128 525V125C128 40 120 34 38 28V0H303Z"/></g></svg>-</span>Expandable Ideal Topological Spaces

Boonpok, Chawalit

doi:https://doi.org/10.1155/2021/9272335

Journal of Mathematics

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Abstract Introduction Preliminaries Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 9272335 | https://doi.org/10.1155/2021/9272335

Semi--Expandable Ideal Topological Spaces

Chawalit Boonpok¹

Academic Editor: Carmelo Antonio Finocchiaro

Received18 Aug 2021

Accepted29 Oct 2021

Published06 Dec 2021

Abstract

Our purpose is to introduce the notion of semi--expandable ideal topological spaces. Some properties of semi--locally finite collections are investigated. In particular, several characterizations of semi--expandable ideal topological spaces are established.

1. Introduction

The concept of expandable spaces was first introduced by Krajewski [1]. Moreover, Krajewski investigated the property of expanding locally finite collection to open finite collection and obtained some results relating this property to certain topological covering properties. Smith et al. [2] introduced various generalizations of the concept of expandability and investigated several characterizations of expandability properties in terms of open covers. Al-Zoubi [3] introduced the concept of -expandable spaces as a variation of expandable spaces and showed that an extremally disconnected semiregular space is -expandable if and only if it is expandable. Jiang and Sun [4] proved that every -space is expandable and discussed a characterization of -expandability for extremally disconnected spaces. Al-Zoubi [5] introduced the class of -paracompact spaces as a generalization of paracompact spaces and investigated the relationships between -paracompact spaces and other well-known spaces. Li and Song [6] introduced and studied -expandable spaces which are a weaker form than -paracompact spaces and showed that -expandability is equivalent to -expandability for extremally disconnected semiregular spaces. Kuratowski [7] and Vaidyanathaswamy [8] introduced and studied the concept of ideal topological spaces. Janković and Hamlett [9] developed the study in logical, systematic fashion and offered some new results, improvements of known results, and some applications. In 2002, Hatir and Noiri [10] introduced the notions of semi--open sets, --open sets, and --open sets via idealization and using these sets obtained new decomposition of continuity. In 2005, Hatir and Noiri [11] investigated some properties of semi--open sets and semi--continuous functions defined in [10] and introduced new functions via ideals, namely, semi--open functions and semi--closed functions. Açikgöz et al. [12] introduced the notion of -submaximal ideal topological spaces and proved that every submaximal space is an -submaximal ideal topological space. In 2009, Ekici and Noiri [13] introduced the notion of -extremally disconnected ideal topological spaces and showed that -extremally disconnectedness and extremally disconnectedness are equivalent to a codense ideal. In 2010, Ekici and Noiri [14] investigated several characterizations of -submaximal ideal topological spaces and proved that semi--open sets and -sets are equivalent to -submaximality and -extremally disconnectedness. In 2012, Ekici and Noiri [15] introduced the notion of -hyperconnected ideal topological spaces and investigated some properties of -hyperconnected ideal topological spaces by utilizing --open sets and the --closure operator. In [16], the author investigated further characterizations of -hyperconnected ideal topological spaces and studied the concept of --irreducible ideal topological spaces.

This paper is organized as follows: in Section 3, we introduce the concept of semi--locally finite collections. Moreover, some properties of semi--locally finite collections are discussed. In Section 4, we introduce the concept of semi--expandable ideal topological spaces. In particular, some characterizations of semi--expandable ideal topological spaces are investigated.

2. Preliminaries

We begin with some definitions and known results which will be used throughout this paper. In the present paper, spaces and (or simply and ) always mean topological spaces on which no separation axioms are assumed unless explicitly stated. In a topological space , the closure and the interior of any subset of will be denoted by and , respectively. A nonempty collection of subsets of a set is said to be an ideal on if satisfies the following two properties: (i) and ; (ii) and . For a topological space with an ideal on , a set operator where is the set of all subsets of , called a local function [7] of with respect to , and is defined as follows: for , where

A Kuratowski closure operator for a topology , which is called the -topology and is finer than , is defined by [9]. We shall simply write for and for . A basis for can be described as follows: . However, is not always a topology [9]. A subset of an ideal topological space is called -closed (-closed) [9] if . The interior of a subset in is denoted by .

A subset of an ideal topological space is called semi--open [10] (resp., --open [15]) if (resp., ). By (resp., ), we denote the family of all semi--open (resp., --open) sets of an ideal topological space . The complement of a semi--open (resp., --open) set is called semi--closed [11] (resp., --closed [15]).

Lemma 1 (see [11]). Let be an ideal topological space and subsets of .(1)If for each , then .(2)If and , then .

The semi--closure (resp., --closure) of a subset of an ideal topological space , denoted by (resp., ), is defined by the intersection of all semi--closed (resp., --closed) sets of containing [15].

Lemma 2 (see [15]). For a subset of an ideal topological space , the following properties hold:(1).(2).

3. Semi--Locally Finite Collections

Recall that a collection of subsets of a topological space is said to be locally finite [17] if, for each , there exists an open set of containing and intersects at most for finitely many .

Definition 1. A collection of subsets of an ideal topological space is said to be semi--locally finite if, for each , there exists a semi--open set of containing and intersects at most for finitely many .

Definition 2. An ideal topological space is said to be semi--regular if, for each semi--closed set and each point , there exist disjoint semi--open sets and such that and .

Lemma 3. An ideal topological space is semi--regular if and only if, for each semi--open set and for each , there exists a semi--open set such that .

Lemma 4 (see [11]). A subset of an ideal topological space is semi--open if and only if there exists such that .

Definition 3 (see [13]). An ideal topological space is said to be -extremally disconnected if the -closure of every open subset of is open.

Lemma 5. For an ideal topological space , the following properties are equivalent:(1) is -extremally disconnected.(2) for each .(3) is -closed for each .

Proof. : Suppose that is a -extremally disconnected space. Let be a semi--open set. Since is semi--open, and, by Lemma 2 (2), : This is obvious. : Let be an open set. Then, is --open. By (3) and Lemma 2 (2),and hence is open. Thus, is -extremally disconnected.

Theorem 1. Let be a -extremally disconnected semi--regular space. Then, the collection of subsets of is semi--locally finite if and only if is locally finite.

Proof. We need to show only necessity. Suppose that is semi--locally finite. For each , there exists a semi--open set containing , and intersects at most finitely many members of . Since is semi--regular, there exists a semi--open set such that . Since is semi--open, by Lemma 4, there exists an open set such that . Since is -extremally disconnected, by Lemma 5, is an open set containing . Since , intersects at most finitely many members of . Thus, is locally finite.

Theorem 2. Let be a collection of subsets of an ideal topological space . Then,(1) is semi--locally finite if and only if is semi--locally finite.(2)If is semi--locally finite, then .(3)If is semi--locally finite and for each , thenis also semi--locally finite.

Proof. The proof is obvious.

Definition 4 (see [18]). A subset of an ideal topological space is called -dense if .

Definition 5 (see [12]). An ideal topological space is called -submaximal if every -dense subset of is open.

Theorem 3. Let be an -submaximal -extremally disconnected space. Then, every semi--locally finite collection of subsets of is locally finite.

Proof. It follows from Corollary 17 of [14].

Definition 6 (see [11]). A function is said to be -irresolute if is semi--open in for every semi--open set of .

Theorem 4. Let be an -irresolute function. If is a semi--locally finite collection in , thenis a semi--locally finite collection in .

Proof. The proof is obvious.

Definition 7. An ideal topological space is said to be semi--compact if every cover of by semi--open sets has a finite subcover.

Definition 8. A function is said to be semi--closed if is semi--closed in for every semi--closed set of .

Lemma 6. A function is semi--closed if and only if, for each and every semi--open set in which contains , there exists a semi--open subset of such that and .

Proof. Suppose that is semi--closed. Let and be any semi--open set in such that . Put . Then, is semi--open in such that and .
Conversely, let be any semi--closed subset of . For each , then . Therefore, there exists a semi--open subset of such that and . Put . Then, is semi--open in such that and . Thus, is semi--closed in .

Theorem 5. Let be a semi--closed function such that is semi--compact in . If is a semi--locally finite collection in , then is a semi--locally finite collection in .

Proof. For each , there exists a semi--open set in containing such that intersects at most finitely many members of . Thus, is a semi--open cover of and so there exist a finite number of points of such that . is semi--closed, so by Lemma 6 there exists a semi--open set of containing such that . Thus, intersects at most finitely many members of and hence is semi--locally finite in .

4. Semi-I-Expandable Ideal Topological Spaces

Recall that a topological space is said to be expandable [1] if, for every locally finite collection of subsets of , there exists a locally finite collection of open subsets of such that for each .

Definition 9. An ideal topological space is said to be semi--expandable (resp., -semi--expandable) if, for every semi--locally finite collection (resp., ) of subsets of , there exists a locally finite collection of open subsets of such that for each .

Now, we have the following characterizations of semi--expandable ideal topological spaces in terms of coverings.

Theorem 6. An ideal topological space is semi--expandable if every semi--open cover of has a locally finite open refinement.

Proof. Let be a semi--locally finite collection of semi--closed subsets of . Let be any finite subset of and let , . Then, is semi--open for each and intersects at most finitely many members of for each . The family is a cover of . For each , there exists a semi--open subset of such that intersects at most many members of , say . By assumption has a locally finite open refinement . Put for each . Then, and is open for each . Now, it suffices to show that is locally finite. Let . Then, there exists an open set which contains and intersects only finite many members of . Thus, if and only if and for some . Since is a refinement of , is contained in some which intersects only finite many . Thus, is locally finite. This shows that is semi--expandable.

Theorem 7. Let be a -extremally disconnected space. Then, is semi--expandable if every semi--open cover of has a locally finite semi--open refinement .

Proof. Let be a semi--locally finite collection of semi--closed subsets of . As in the proof of Theorem 6, we construct a locally finite semi--open collection of subsets of such that for each . Since is semi--open in for each , we choose such that . Since is -extremally disconnected, the collection is open locally finite in and for each . Thus, is semi--expandable.

Theorem 8. Let be a -extremally disconnected space. Then, is -semi--expandable if and only if every countable semi--open cover of has a locally finite semi--open refinement .

Proof. It follows from Theorem 7.
Conversely, let be -semi--expandable and let be a countable semi--open cover of . For each , put . Then,is an increasing semi--open cover of , and hence the collectionis semi--locally finite in . Therefore, there exists a locally finite collection of open subsets of such that for each . Now, for each , put . Then, by Lemma 1, is semi--open in and for each . Finally, since is locally finite, it is easy to show that the collection is a locally finite refinement of .

Theorem 9. Let be an -irresolute -closed surjection and be compact for each . If is semi--expandable, then is semi--expandable.

Proof. Let be a semi--locally finite collection in . By Theorem 4, the collection is semi--locally finite in , and so there exists a locally finite collection of open subsets of such that for each . Put for each . It is easy to see that is open and for each . Finally, we show that the collection of subsets of is locally finite. Let . For each , there exists an open set containing such that intersects at most finitely many members of . Therefore, the collection is an open cover of , and so there exist a finite number of points of such that . Since is -closed, there exists an open set containing and . Since for each and is locally finite in , intersects at most finitely many members of which means that is locally finite in . Thus, is semi--expandable.

Theorem 10. Let be a continuous semi--closed surjection and let be semi--compact in for each . If is semi--expandable, then is semi--expandable.

Proof. Let be a semi--locally finite collection of subsets of . By Theorem 5, is a semi--locally finite collection in . Then, there exists a locally finite collection of open subsets of such that for each . Then, and is an open locally finite collection in . Thus, is semi--expandable.

Theorem 11. For an ideal topological space , the following properties are equivalent:(1) is semi--expandable.(2) is expandable and every semi--locally finite collection of is locally finite.

Proof. : Let be semi--expandable and a semi--locally finite collection of subsets of . Then, there exists a locally finite collection of open subsets of such that for each . Observe that local finiteness of implies that the family must have been locally finite. : It follows directly from the definition of semi--expandability.

Theorem 12. Let be an ideal topological space. If, for every semi--locally finite collection , there exists a locally finite open cover of such that each element of meets only finitely many elements of , then is semi--expandable.

Proof. Let be a semi--locally finite collection of subsets of and the locally finite open cover of such that each element of intersects only finitely many elements of . Put , . Clearly, and is open for each . We claim thatis locally finite. For each , since is locally finite, there exists an open neighborhood of which meets only finitely many members of . Then,if and only if and for some . Since meets only finitely many members of , intersects only finite many members of . Thus, is locally finite and hence is semi--expandable.

Corollary 1. Let be a -extremally disconnected space. If, for every semi--locally finite collection , there exists a locally finite semi--open cover of such that each element of meets only finitely many elements of , then is semi--expandable.

Definition 10. A collection of subsets of an ideal topological space is said to be -semi--locally finite if , where each is semi--locally finite.

Theorem 13. Let be a -extremally disconnected space. Then, is semi--expandable if and only if every semi--open cover of with a -semi--locally finite refinement has a locally finite semi--open refinement.

Proof. Suppose that every semi--open cover of with a -semi--locally finite refinement has a locally finite semi--open refinement. Let be a semi--locally finite collection of semi--closed subsets of . Definefor each , , . Then, is a semi--open cover of . For each , intersects only finitely many members of . Now define , for each and . For each , let . Since is semi--locally finite, is a -semi--locally refinement of . Hence, has a locally finite semi--open refinement . Thus, each element of meets only finitely many members of ; by Corollary 1, is semi--expandable.
Conversely, let be a semi--expandable and -extremally disconnected space. Let be a semi--open cover of with a -semi--locally finite refinement , where is semi--locally finite for each . We will prove that has a locally finite semi--open refinement. Since is semi--expandable, for each , there exists a locally finite open collection such that for each . Since is a refinement of , for any , there exists such that . For each and , let , . It follows from Lemma 1 that is a -locally finite semi--open refinement of . Put for each . By the definition of semi--open sets, is semi--open, and hence is a countable semi--open cover of . Since is semi--expandable, is -semi--expandable. By Theorem 8, there exists a locally finite semi--open refinement , and we may assume that . Since is semi--open for each , there exists an open set such that . Since is -extremally disconnected, is a locally finite collection of open subsets of . Let . It is easy to check that is a locally finite semi--open refinement of .

Lemma 7 (see [19]). Let be an ideal topological space, and . If is semi--open in , then is semi--open in .

Theorem 14. Let be a semi--expandable ideal topological space. If is clopen subset of , then is semi--expandable.

Proof. Let be a clopen subset of a semi--expandable space . Let be a semi--locally finite collection of subsets of . Since is clopen in , is semi--locally finite in for each . Then, either or . If , then there exists containing such that intersects at most finitely many members of . Since is open, , by Lemma 7, and hence is a semi--locally finite collection in . If , then is semi--open in containing which intersects no member of . Thus, is a semi--locally finite collection of the semi--expandable ideal topological space , so there exists a locally finite collection of open subsets of , say such that for each . Now, consider . Then, is a locally finite collection of open subsets of such that for each . Thus, is semi--expandable.

Recall that a topological space is said to be paracompact [20] if every open cover of has a locally finite open refinement.

Definition 11. An ideal topological space is called semi--paracompact if every open cover of has a locally finite semi--open refinement.

Theorem 15. Let be a -semi--expandable -extremally disconnected space. Then, is semi--paracompact if every open cover of has a -locally finite semi--open refinement.

Proof. Let be an open cover and be a locally finite collection of semi--open subsets of . Let for each . Then, is semi--open by the definition of semi--open sets. is a countable semi--open cover of and, hence, by Theorem 8 has a locally finite semi--open refinement , and we may assume that for each . Since is semi--open for each , there exists an open set such that . Thus, is a locally finite collection of open subsets since is a -extremally disconnected space. Therefore, is a locally finite semi--open refinement of by Lemma 1 (2). Thus, is semi--paracompact.

Definition 12. A function is said to be semi--open if is semi--open in for every semi--open set of .

Theorem 16. Let be a continuous, semi--open, and closed surjection such that is compact for each . If is semi--paracompact, then is semi--paracompact.

Proof. Let be an open cover of . Then,is an open cover of the semi--paracompact space and so has a locally finite semi--open refinement, say . Since is semi--open, the collection is a semi--open refinement of . Finally, we shall show that the collection is locally finite in . Let . For each , there exists an open set containing such that intersects at most finitely many members of . The collectionis an open cover of , and therefore there exists a finite subset of such that . Since is closed, there exists an open set containing such that . Then, intersects at most finitely many members of . Therefore, intersects at most finitely many members of . Thus, is locally finite in .

5. Semi--Paracompact Subsets

We begin this section by introducing the concept of semi--paracompact subsets. In particular, some properties of semi--paracompact subsets are discussed.

Definition 13. A subset of an ideal topological space is said to be semi--paracompact set if every cover by open subsets of has a locally finite semi--open refinement in .
Recall that a subset of an ideal topological space is called -closed [21] if , whenever is open and .

Lemma 8 (see [22]). For a subset of an ideal topological space , the following properties are equivalent:(a) is -closed.(b) whenever and is open in .(c)For all , .(d) contains no nonempty closed set.(e) contains no nonempty closed set.

Theorem 17. Every -closed subset of a semi--paracompact space is semi--paracompact.

Proof. Let be a semi--paracompact space and let be a -closed subset of . Let be any cover of by open subsets of . Since and is -closed, by Lemma 8, . For each , there exists an open set of containing such that . Now, put . Then, is an open cover of the semi--paracompact space . Let be a locally finite semi--open refinement of . Then, for each , either for some or for some . Put . Then, is a locally finite semi--open refinement of and . Thus, is semi--paracompact.

Theorem 18. Every open semi--paracompact subset of an ideal topological space is semi--paracompact.

Proof. Let be an open semi--paracompact subset of an ideal topological space . Let be any open cover of by open subsets of the subspace . Since is open, is a cover of by open subsets of and so has a locally finite semi--open refinement of in . Then, is a locally finite semi--open refinement of in and the result follows.

Theorem 19. Let be a clopen subspace of an ideal topological space . Then, is semi--paracompact if and only if it is semi--paracompact.

Proof. It follows from Theorem 18.
Conversely, let be any open cover of by open subsets of . Then, is an open cover of the semi--paracompact subspace and so has a locally finite semi--open refinement in . By Lemma 7, for every . To show that is locally finite in , let . If , then there exists containing such that intersects at most finitely many members of . Otherwise, is an open set containing which intersects no member of . Therefore, W is locally finite in such that . Thus, is semi--paracompact.

Corollary 2. Every clopen subspace of a semi--paracompact space is semi--paracompact.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there are no conflicts of interest.

Acknowledgments

This research project was financially supported by Mahasarakham University.

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Copyright © 2021 Chawalit Boonpok. 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.

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