#### 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 , then**is 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.