# Semi-I-Expandable Ideal Topological Spaces

Authors: Chawalit Boonpok
Journal: Journal of Mathematics
 (2021)

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## Abstract

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


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## Body

## 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 s-expandable spaces as a variation of expandable spaces and showed that an extremally disconnected semiregular space is s-expandable if and only if it is expandable. Jiang and Sun [4] proved that every T1wN-space is expandable and discussed a characterization of s-expandability for extremally disconnected spaces. Al-Zoubi [5] introduced the class of S-paracompact spaces as a generalization of paracompact spaces and investigated the relationships between S-paracompact spaces and other well-known spaces. Li and Song [6] introduced and studied S-expandable spaces which are a weaker form than S-paracompact spaces and showed that s-expandability is equivalent to S-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-I-open sets, α-I-open sets, and β-I-open sets via idealization and using these sets obtained new decomposition of continuity. In 2005, Hatir and Noiri [11] investigated some properties of semi-I-open sets and semi-I-continuous functions defined in [10] and introduced new functions via ideals, namely, semi-I-open functions and semi-I-closed functions. Açikgöz et al. [12] introduced the notion of I-submaximal ideal topological spaces and proved that every submaximal space is an I-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 I-submaximal ideal topological spaces and proved that semi-I-open sets and ABI-sets are equivalent to I-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 semi∗-I-open sets and the semi∗-I-closure operator. In [16], the author investigated further characterizations of ∗-hyperconnected ideal topological spaces and studied the concept of θ-I-irreducible ideal topological spaces.This paper is organized as follows: in Section3, we introduce the concept of semi-I-locally finite collections. Moreover, some properties of semi-I-locally finite collections are discussed. In Section 4, we introduce the concept of semi-I-expandable ideal topological spaces. In particular, some characterizations of semi-I-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, spacesX,τ and Y,σ (or simply X and Y) always mean topological spaces on which no separation axioms are assumed unless explicitly stated. In a topological space X,τ, the closure and the interior of any subset A of X will be denoted by ClA and IntA, respectively. A nonempty collection I of subsets of a set X is said to be an ideal on X if I satisfies the following two properties: (i) A∈I and B⊆A⟹B∈I; (ii) A∈I and B∈I⟹A∪B∈I. For a topological space X,τ with an ideal I on X, a set operator .∗:PX⟶PX where PX is the set of all subsets of X, called a local function [7] of A with respect to I, and τ is defined as follows: for A⊆X, A∗I,τ=x∈X|G∩A∉Ifor everyG∈τx where(1)τx=G∈τx∈G.A Kuratowski closure operatorCl∗. for a topology τ∗I,τ, which is called the ∗-topology and is finer than τ, is defined by Cl∗A=A∪A∗ [9]. We shall simply write A∗ for A∗I,τ and τ∗ for τ∗I,τ. A basis BI,τ for τ∗ can be described as follows: BI,τ=V−I′|V∈τandI′∈I. However, BI,τ is not always a topology [9]. A subset A of an ideal topological space X,τ,I is called ∗-closed (τ∗-closed) [9] if A∗⊆A. The interior of a subset A in X,τ∗I,τ is denoted by Int∗A.A subsetA of an ideal topological space X,τ,I is called semi-I-open [10] (resp., semi∗-I-open [15]) if A⊆Cl∗IntA (resp., A⊆Cl∗IntA). By sIOX,τ (resp., s∗IOX,τ), we denote the family of all semi-I-open (resp., semi∗-I-open) sets of an ideal topological space X,τ,I. The complement of a semi-I-open (resp., semi∗-I-open) set is called semi-I-closed [11] (resp., semi∗-I-closed [15]).Lemma 1 (see [11]).
LetX,τ,I be an ideal topological space and A,B subsets of X.(1)
IfUα∈sIOX,τ for each α∈Δ, then ∪Uα:α∈Δ∈sIOX,τ.(2)
IfA∈sIOX,τ and B∈τ, then A∩B∈sIOX,τ.The semi-I-closure (resp., semi∗-I-closure) of a subset A of an ideal topological space X,τ,I, denoted by sClIA (resp., s∗ClIA), is defined by the intersection of all semi-I-closed (resp., semi∗-I-closed) sets of X containing A [15].Lemma 2 (see [15]).
For a subsetA of an ideal topological space X,τ,I, the following properties hold:(1)
sClIA=A∪Int∗ClA.(2)
s∗ClIA=A∪IntCl∗A.


## 3. Semi-I-Locally Finite Collections

Recall that a collectionC=Cγ|γ∈Γ of subsets of a topological space X,τ is said to be locally finite [17] if, for each x∈X, there exists an open set U of X containing x and U intersects Cγ at most for finitely many γ.Definition 1.
A collectionC=Cγ|γ∈Γ of subsets of an ideal topological space X,τ,I is said to be semi-I-locally finite if, for each x∈X, there exists a semi-I-open set U of X containing x and U intersects Cγ at most for finitely many γ.Definition 2.
An ideal topological spaceX,τ,I is said to be semi-I-regular if, for each semi-I-closed set F and each point x∉F, there exist disjoint semi-I-open sets U and V such that x∈U and F⊆V.Lemma 3.
An ideal topological spaceX,τ,I is semi-I-regular if and only if, for each semi-I-open set U and for each x∈U, there exists a semi-I-open set V such that x∈V⊆sClIV⊆U.Lemma 4 (see [11]).
A subsetA of an ideal topological space X,τ,I is semi-I-open if and only if there exists U∈τ such that U⊆A⊆Cl∗U.Definition 3 (see [13]).
An ideal topological spaceX,τ,I is said to be ∗-extremally disconnected if the ∗-closure of every open subset U of X is open.Lemma 5.
For an ideal topological spaceX,τ,I, the following properties are equivalent:(1)
X,τ,I is ∗-extremally disconnected.(2)
s∗ClIU=Cl∗U for each U∈sIOX,τ.(3)
s∗ClIU is ∗-closed for each U∈sIOX,τ.Proof.
1⟹2: Suppose that X,τ,I is a ∗-extremally disconnected space. Let U be a semi-I-open set. Since U is semi-I-open, Cl∗U=Cl∗IntU and, by Lemma 2 (2),(2)s∗ClIU=U∪IntCl∗U=U∪IntCl∗IntU=U∪Cl∗IntU=Cl∗IntU=Cl∗U.2⟹3: This is obvious.3⟹1: Let U be an open set. Then, U is semi-I-open. By (3) and Lemma 2 (2),(3)IntCl∗U=U∪IntCl∗U=s∗ClIU=Cl∗s∗ClIU=Cl∗U∪IntCl∗U=Cl∗IntCl∗U=Cl∗U,and hence Cl∗U is open. Thus, X,τ,I is ∗-extremally disconnected.Theorem 1.
LetX,τ,I be a ∗-extremally disconnected semi-I-regular space. Then, the collection C=Cγ|γ∈Γ of subsets of X is semi-I-locally finite if and only if C is locally finite.Proof.
We need to show only necessity. Suppose thatC is semi-I-locally finite. For each x∈X, there exists a semi-I-open set U containing x, and U intersects at most finitely many members of C. Since X,τ,I is semi-I-regular, there exists a semi-I-open set V such that x∈V⊆sClIV⊆U. Since V is semi-I-open, by Lemma 4, there exists an open set W such that W⊆V⊆Cl∗W. Since X,τ,I is ∗-extremally disconnected, by Lemma 5, Cl∗W=Cl∗V=s∗ClIV is an open set containing x. Since s∗ClIV⊆sClIV, Cl∗W intersects at most finitely many members of C. Thus, C is locally finite.Theorem 2.
LetC=Cγ|γ∈Γ be a collection of subsets of an ideal topological space X,τ,I. Then,(1)
C is semi-I-locally finite if and only if sClICγ|γ∈Γ is semi-I-locally finite.(2)
IfC is semi-I-locally finite, then ∪γ∈ΓsClICγ=sClI∪γ∈ΓCγ.(3)
IfC is semi-I-locally finite and Dγ⊆Cγ for each γ∈Γ, then(4)D=Dγ|γ∈Γ,is also semi-I-locally finite.Proof.
The proof is obvious.Definition 4 (see [18]).
A subsetD of an ideal topological space X,τ,I is called ∗-dense if Cl∗D=X.Definition 5 (see [12]).
An ideal topological spaceX,τ,I is called I-submaximal if every ∗-dense subset of X is open.Theorem 3.
LetX,τ,I be an I-submaximal ∗-extremally disconnected space. Then, every semi-I-locally finite collection C=Cγ|γ∈Γ of subsets of X is locally finite.Proof.
It follows from Corollary 17 of [14].Definition 6 (see [11]).
A functionf:X,τ,I⟶Y,σ,J is said to be I-irresolute if f−1V is semi-I-open in X,τ,I for every semi-J-open set V of Y,σ,J.Theorem 4.
Letf:X,τ,I⟶Y,σ,J be an I-irresolute function. If C=Cγ|γ∈Γ is a semi-J-locally finite collection in Y,σ,J, then(5)f−1Cγ|γ∈Γ,is a semi-I-locally finite collection in X,τ,I.Proof.
The proof is obvious.Definition 7.
An ideal topological spaceX,τ,I is said to be semi-I-compact if every cover of X by semi-I-open sets has a finite subcover.Definition 8.
A functionf:X,τ,I⟶Y,σ,J is said to be semi-I,J-closed if fF is semi-J-closed in Y,σ,J for every semi-I-closed set F of X,τ,I.Lemma 6.
A functionf:X,τ,I⟶Y,σ,J is semi-I,J-closed if and only if, for each y∈Y and every semi-I-open set U in X,τ,I which contains f−1y, there exists a semi-J-open subset V of Y,σ,J such that y∈V and f−1V⊆U.Proof.
Suppose thatf is semi-I,J-closed. Let y∈Y and U be any semi-I-open set in X,τ,I such that f−1y⊆U. Put V=Y−fX−U. Then, V is semi-J-open in Y,σ,J such that y∈V and f−1V⊆U.
Conversely, letF be any semi-I-closed subset of X. For each y∈Y−fF, then f−1y⊆X−F=U. Therefore, there exists a semi-J-open subset Vy of Y,σ,J such that y∈Vy and f−1Vy⊆U. Put V=∪Vy|y∈Y−fF. Then, V is semi-J-open in Y,σ,J such that y∈V and f−1V⊆U. Thus, fF is semi-J-closed in Y,σ,J.Theorem 5.
Letf:X,τ,I⟶Y,σ,J be a semi-I,J-closed function such that f−1y is semi-I-compact in X,τ,I. If C=Cγ|γ∈Γ is a semi-I-locally finite collection in X,τ,I, then fC=fCγ|γ∈Γ is a semi-J-locally finite collection in Y,σ,J.Proof.
For eachx∈f−1y, there exists a semi-I-open set Ux in X,τ,I containing x such that Ux intersects at most finitely many members of C. Thus, Ux|x∈f−1y is a semi-I-open cover of f−1y and so there exist a finite number of points x1,x2,…,xn of f−1y such that f−1y⊆∪i=1nUxi=U. f is semi-I,J-closed, so by Lemma 6 there exists a semi-J-open set V of Y containing y such that f−1V⊆U. Thus, V intersects at most finitely many members of fC and hence fC is semi-J-locally finite in Y,σ,J.


## 4. Semi-I-Expandable Ideal Topological Spaces

Recall that a topological spaceX,τ is said to be expandable [1] if, for every locally finite collection C=Cγ|γ∈Γ of subsets of X, there exists a locally finite collection G=Gγ|γ∈Γ of open subsets of X such that Cγ⊆Gγ for each γ∈Γ.Definition 9.
An ideal topological spaceX,τ,I is said to be semi-I-expandable (resp., ω0-semi-I-expandable) if, for every semi-I-locally finite collection C=Cγ|γ∈Γ (resp., Γ≤ω0) of subsets of X, there exists a locally finite collection G=Gγ|γ∈Γ of open subsets of X such that Cγ⊆Gγ for each γ∈Γ.Now, we have the following characterizations of semi-I-expandable ideal topological spaces in terms of coverings.Theorem 6.
An ideal topological spaceX,τ,I is semi-I-expandable if every semi-I-open cover of X has a locally finite open refinement.Proof.
LetC=Cγ|γ∈Γ be a semi-I-locally finite collection of semi-I-closed subsets of X. Let Γ0 be any finite subset of Γ and let Vλ=X−∪Cγ|γ≠λ, λ∈Γ0. Then, Vλ is semi-I-open for each λ∈Γ0 and Vλ intersects at most finitely many members of C for each λ∈Γ0. The family V=Vλ|λ∈Γ0 is a cover of X. For each x∈X, there exists a semi-I-open subset Vx of X such that Vx intersects at most many members of C, say Cγ1,Cγ2,…,Cγn. By assumption V has a locally finite open refinement W=Wβ|β∈Ω. Put Uγ=∪W′∈W|W′∩Cγ≠∅ for each γ∈Γ. Then, Cγ⊆Uγ and Uγ is open for each γ∈Γ. Now, it suffices to show that U=Uγ|γ∈Γ is locally finite. Let x∈X. Then, there exists an open set Gx which contains x and intersects only finite many members of W. Thus, Gx∩Uγ≠∅ if and only if Gx∩Wβ≠∅ and Wβ∩Cγ≠∅ for some β∈Ω. Since W is a refinement of V, Wβ is contained in some Vλ which intersects only finite many Cγ. Thus, U is locally finite. This shows that X,τ,I is semi-I-expandable.Theorem 7.
LetX,τ,I be a ∗-extremally disconnected space. Then, X,τ,I is semi-I-expandable if every semi-I-open cover U of X has a locally finite semi-I-open refinement V.Proof.
LetC=Cγ|γ∈Γ be a semi-I-locally finite collection of semi-I-closed subsets of X. As in the proof of Theorem 6, we construct a locally finite semi-I-open collection U=Uγ|γ∈Γ of subsets of X such that Cγ⊆Uγ for each γ∈Γ. Since Uγ is semi-I-open in X,τ,I for each γ∈Γ, we choose Vγ∈τ such that Vγ⊆Uγ⊆Cl∗Vγ. Since X,τ,I is ∗-extremally disconnected, the collection V=Cl∗Vγ|γ∈Γ is open locally finite in X,τ,I and Cγ⊆Uγ⊆Cl∗Vγ for each γ∈Γ. Thus, X,τ,I is semi-I-expandable.Theorem 8.
LetX,τ,I be a ∗-extremally disconnected space. Then, X,τ,I is ω0-semi-I-expandable if and only if every countable semi-I-open cover U of X has a locally finite semi-I-open refinement V.Proof.
It follows from Theorem7.
Conversely, letX,τ,I be ω0-semi-I-expandable and let U=Un|n∈ℕ be a countable semi-I-open cover of X. For each n, put Wn=∪n0≤nUn0. Then,(6)W=Wn|n∈ℕis an increasing semi-I-open cover of X, and hence the collection(7)C=Cn=X−Wn|n∈ℕis semi-I-locally finite in X,τ,I. Therefore, there exists a locally finite collection G=Gn|n∈ℕ of open subsets of X,τ,I such that Cn⊆Gn for each n. Now, for each n∈ℕ, put Vn=Un−∪n0<nX−Gn0. Then, by Lemma 1, Vn is semi-I-open in X,τ,I and Vn⊆Un for each n. Finally, since G is locally finite, it is easy to show that the collection V=Vn|n∈ℕ is a locally finite refinement of U.Theorem 9.
Letf:X,τ,I⟶Y,σ,J be an I-irresolute I,J-closed surjection and f−1y be compact for each y∈Y. If X,τ,I is semi-I-expandable, then Y,σ,J is semi-J-expandable.Proof.
LetC=Cγ|γ∈Γ be a semi-J-locally finite collection in Y,σ,J. By Theorem 4, the collection f−1C=f−1Cγ|γ∈Γ is semi-I-locally finite in X,τ,I, and so there exists a locally finite collection G=Gγ|γ∈Γ of open subsets of X such that f−1Cγ⊆Gγ for each γ∈Γ. Put Vγ=Y−fX−Gγ for each γ∈Γ. It is easy to see that Vγ is open and Fγ⊆Vγ for each γ∈Γ. Finally, we show that the collection V=Vγ|γ∈Γ of subsets of Y is locally finite. Let y∈Y. For each x∈f−1y, there exists an open set Ux containing x such that Ux intersects at most finitely many members of G. Therefore, the collection Ux|x∈f−1y is an open cover of f−1y, and so there exist a finite number of points x1,x2,…,xn of f−1y such that f−1y⊆∪i=1nUxi=U. Since f is I,J-closed, there exists an open set Vy containing y and f−1V⊆U. Since f−1Vγ⊆Gγ for each γ∈Γ and G is locally finite in X,τ,I, Vy intersects at most finitely many members of V which means that V is locally finite in Y,σ,J. Thus, Y,σ,J is semi-J-expandable.Theorem 10.
Letf:X,τ,I⟶Y,σ,J be a continuous semi-I,J-closed surjection and let f−1y be semi-I-compact in X,τ,I for each y∈Y. If Y,σ,J is semi-J-expandable, then X,τ,I is semi-I-expandable.Proof.
LetC=Cγ∈γ be a semi-I-locally finite collection of subsets of X,τ,I. By Theorem 5, fCγ|γ∈Γ is a semi-J-locally finite collection in Y,σ,J. Then, there exists a locally finite collection G=Gγ|γ∈Γ of open subsets of Y,σ,J such that fCγ⊆Gγ for each γ∈Γ. Then, Cγ⊆f−1fCγ⊆f−1Gγ and f−1Gγ|γ∈Γ is an open locally finite collection in X,τ,I. Thus, X,τ,I is semi-I-expandable.Theorem 11.
For an ideal topological spaceX,τ,I, the following properties are equivalent:(1)
X,τ,I is semi-I-expandable.(2)
X,τ,I is expandable and every semi-I-locally finite collection of X is locally finite.Proof.
1⟹2: Let X,τ,I be semi-I-expandable and C=Cγ|γ∈Γ a semi-I-locally finite collection of subsets of X. Then, there exists a locally finite collection G=Gγ|γ∈Γ of open subsets of X such that Cγ⊆Gγ for each γ∈Γ. Observe that local finiteness of G implies that the family C must have been locally finite.2⟹1: It follows directly from the definition of semi-I-expandability.Theorem 12.
LetX,τ,I be an ideal topological space. If, for every semi-I-locally finite collection C, there exists a locally finite open cover U of X such that each element of U meets only finitely many elements of C, then X,τ,I is semi-I-expandable.Proof.
LetC=Cγ|γ∈Γ be a semi-I-locally finite collection of subsets of X and U=Uδ|δ∈∇ the locally finite open cover of X such that each element of U intersects only finitely many elements of C. Put Wγ=∪Uδ∈U|Uδ∩Cγ≠∅, γ∈Γ. Clearly, Cγ⊆Wγ and Wγ is open for each γ∈Γ. We claim that(8)W=Wγ|γ∈Γis locally finite. For each x∈X, since U is locally finite, there exists an open neighborhood Nx of x which meets only finitely many members of U. Then,(9)Nx∩Wγ≠∅if and only if Nx∩Uδ≠∅ and Uδ∩Cγ≠∅ for some δ∈∇. Since Uδ meets only finitely many members of C, Nx intersects only finite many members of W. Thus, W is locally finite and hence X,τ,I is semi-I-expandable.Corollary 1.
LetX,τ,I be a ∗-extremally disconnected space. If, for every semi-I-locally finite collection C, there exists a locally finite semi-I-open cover U of X such that each element of U meets only finitely many elements of C, then X,τ,I is semi-I-expandable.Definition 10.
A collectionC of subsets of an ideal topological space X,τ,I is said to be σ-semi-I-locally finite if C=∪n=1∞Cn, where each Cn is semi-I-locally finite.Theorem 13.
LetX,τ,I be a ∗-extremally disconnected space. Then, X,τ,I is semi-I-expandable if and only if every semi-I-open cover of X with a σ-semi-I-locally finite refinement has a locally finite semi-I-open refinement.Proof.
Suppose that every semi-I-open cover of X with a σ-semi-I-locally finite refinement has a locally finite semi-I-open refinement. Let C=Cγ|γ∈Γ be a semi-I-locally finite collection of semi-I-closed subsets of X. Define(10)Bn=Ω⊆Γ|Ω=n,for each n∈ℕ, Un=X−∪γ∉ΩCγ|Ω∈Bn, U0=X−∪γ∈ΓCγ. Then, U=∪n=1∞Un is a semi-I-open cover of X. For each U′∈U, U′ intersects only finitely many members of C. Now define Gn=x∈X|ordx,C=n, for each n∈ℕ and V0=U0. For each n, let Vn=∩γ∈ΩCγ∩Gn|Ω∈Bn. Since C is semi-I-locally finite, V=∪i=0∞Vi is a σ-semi-I-locally refinement of U. Hence, U has a locally finite semi-I-open refinement W. Thus, each element of W meets only finitely many members of C; by Corollary 1, X,τ,I is semi-I-expandable.
Conversely, letX,τ,I be a semi-I-expandable and ∗-extremally disconnected space. Let U=Uλ|λ∈Ω be a semi-I-open cover of X with a σ-semi-I-locally finite refinement L=∪n=1∞Ln, where Ln=Lγ,n|γ∈Γn is semi-I-locally finite for each n. We will prove that U has a locally finite semi-I-open refinement. Since X,τ,I is semi-I-expandable, for each n, there exists a locally finite open collection Gγ,n|γ∈Γn such that Lγ,n⊆Gγ,n for each γ∈Γn. Since L is a refinement of U, for any Lγ,n∈L, there exists λγ,n∈Ω such that Lγ,n⊆Uλγ,n. For each n∈ℕ and γ∈Γn, let Wγ,n=Gγ,n∩Uλγ,n, Wn=Wγ,n|γ∈Γn. It follows from Lemma 1 that W=∪n=1∞Wn is a σ-locally finite semi-I-open refinement of U. Put Sn=∪Wγ,n|γ∈Γn for each n∈ℕ. By the definition of semi-I-open sets, Sn is semi-I-open, and hence Sn|n∈ℕ is a countable semi-I-open cover of X. Since X,τ,I is semi-I-expandable, X,τ,I is ω0-semi-I-expandable. By Theorem 8, there exists a locally finite semi-I-open refinement Vn|n∈ℕ, and we may assume that Vn⊆Sn. Since Vn is semi-I-open for each n, there exists an open set Gn such that Gn⊆Vn⊆Cl∗Gn. Since X,τ,I is ∗-extremally disconnected, Cl∗Gn|n∈ℕ is a locally finite collection of open subsets of X. Let G=Cl∗Gn∩Wγ,n|γ∈Γn,n∈ℕ. It is easy to check that G is a locally finite semi-I-open refinement of U.Lemma 7 (see [19]).
LetX,τ,I be an ideal topological space, A⊆U⊆X and U∈τ. If A is semi-I|U-open in U,τ|U,I|U, then A is semi-I-open in X,τ,I.Theorem 14.
LetX,τ,I be a semi-I-expandable ideal topological space. If U is clopen subset of X,τ,I, then U,τ|U,I|U is semi-I|U-expandable.Proof.
LetU be a clopen subset of a semi-I-expandable space X,τ,I. Let C=Cγ|γ∈Γ be a semi-I-locally finite collection of subsets of U,τ|U,I|U. Since U is clopen in X,τ,I, C is semi-I-locally finite in X,τ,I for each x∈X. Then, either x∈U or x∉U. If x∈U, then there exists V∈sI|UOU,τ|U containing x such that V intersects at most finitely many members of C. Since U is open, V∈sIOX,τ, by Lemma 7, and hence C is a semi-I-locally finite collection in X,τ,I. If x∉U, then X−U is semi-I-open in X,τ,I containing x which intersects no member of C. Thus, C is a semi-I-locally finite collection of the semi-I-expandable ideal topological space X,τ,I, so there exists a locally finite collection of open subsets of X,τ,I, say G=Gγ|γ∈Γ such that Cγ⊆Gγ for each γ∈Γ. Now, consider G0=Gγ∩U|γ∈Γ. Then, G0 is a locally finite collection of open subsets of U such that Cγ⊆U∩Gγ for each γ∈Γ. Thus, U,τ|U,I|U is semi-I|U-expandable.Recall that a topological spaceX,τ is said to be paracompact [20] if every open cover of X has a locally finite open refinement.Definition 11.
An ideal topological spaceX,τ,I is called semi-I-paracompact if every open cover of X has a locally finite semi-I-open refinement.Theorem 15.
LetX,τ,I be a ω0-semi-I-expandable ∗-extremally disconnected space. Then, X,τ,I is semi-I-paracompact if every open cover of X has a σ-locally finite semi-I-open refinement.Proof.
LetU=Uγ|γ∈Γ be an open cover and V=Vγ,n|γ∈Γn be a locally finite collection of semi-I-open subsets of X. Let Wn=∪γ∈ΓnVγ,n for each n∈ℕ. Then, Wn is semi-I-open by the definition of semi-I-open sets. Wn|n∈ℕ is a countable semi-I-open cover of X and, hence, by Theorem 8Wn|n∈ℕ has a locally finite semi-I-open refinement H=Hn|n∈ℕ, and we may assume that Hn⊆Wn for each n∈ℕ. Since Hn is semi-I-open for each n, there exists an open set Gn such that Gn⊆Hn⊆Cl∗Gn. Thus, Cl∗Gn|n∈ℕ is a locally finite collection of open subsets since X,τ,I is a ∗-extremally disconnected space. Therefore, G=Cl∗Gn∩Vγ,n|γ∈Γn,n∈ℕ is a locally finite semi-I-open refinement of U by Lemma 1 (2). Thus, X,τ,I is semi-I-paracompact.Definition 12.
A functionf:X,τ,I⟶Y,σ,J is said to be semi-I,J-open if fG is semi-J-open in Y,σ,J for every semi-I-open set G of X,τ,I.Theorem 16.
Letf:X,τ,I⟶Y,σ,J be a continuous, semi-I,J-open, and closed surjection such that f−1y is compact for each y∈Y. If X,τ,I is semi-I-paracompact, then Y,σ,J is semi-J-paracompact.Proof.
LetU=Uγ|γ∈Γ be an open cover of Y,σ,J. Then,(11)f−1U=f−1Uγ|γ∈Γis an open cover of the semi-I-paracompact space X,τ,I and so f−1U has a locally finite semi-I-open refinement, say V=Vα|α∈∇. Since f is semi-I,J-open, the collection fV=fVα|α∈∇ is a semi-J-open refinement of U. Finally, we shall show that the collection fV is locally finite in Y,σ,J. Let y∈Y. For each x∈f−1y, there exists an open set Gx containing x such that Gx intersects at most finitely many members of V. The collection(12)Gx|x∈f−1yis an open cover of f−1y, and therefore there exists a finite subset K0 of f−1y such that f−1y⊆∪x∈K0Gx. Since f is closed, there exists an open set Wy containing y such that f−1Wy⊆∪x∈K0Gx. Then, f−1Wy intersects at most finitely many members of V. Therefore, Wy intersects at most finitely many members of fV. Thus, fV is locally finite in Y,σ,J.


## 5. Semi-I∗-Paracompact Subsets

We begin this section by introducing the concept of semi-I∗-paracompact subsets. In particular, some properties of semi-I∗-paracompact subsets are discussed.Definition 13.
A subsetA of an ideal topological space X,τ,I is said to be semi-I∗-paracompact set if every cover A by open subsets of X,τ,I has a locally finite semi-I-open refinement in X,τ,I.
Recall that a subsetA of an ideal topological space X,τ,I is called Ig-closed [21] if A∗⊆U, whenever U is open and A⊆U.Lemma 8 (see [22]).
For a subsetA of an ideal topological space X,τ,I, the following properties are equivalent:(a)
A is Ig-closed.(b)
Cl∗A⊆U whenever A⊆U and U is open in X.(c)
For allx∈Cl∗A, Clx∩A≠∅.(d)
Cl∗A−A contains no nonempty closed set.(e)
A∗−A contains no nonempty closed set.Theorem 17.
EveryIg-closed subset of a semi-I-paracompact space is semi-I∗-paracompact.Proof.
LetX,τ,I be a semi-I-paracompact space and let A be a Ig-closed subset of X. Let U=Uγ|γ∈Γ be any cover of A by open subsets of X. Since A⊆∪γ∈ΓUγ and A is Ig-closed, by Lemma 8, Cl∗A⊆∪γ∈ΓUγ. For each x∉Cl∗A, there exists an open set Gx of X containing x such that Gx∩A=∅. Now, put U0=Uγ|γ∈Γ∪Gx|x∉Cl∗A. Then, U0 is an open cover of the semi-I-paracompact space X,τ,I. Let V=Vλ|λ∈∇ be a locally finite semi-I-open refinement of U0. Then, for each λ∈∇, either Vλ⊆Uγλ for some γλ or Vλ⊆Gxλ for some xλ. Put ∇0=λ∈∇|Vλ⊆Uγλ. Then, V0=Vλ|λ∈∇0 is a locally finite semi-I-open refinement of U and A⊆∪λ∈∇0Vλ. Thus, A is semi-I∗-paracompact.Theorem 18.
Every open semi-I∗-paracompact subset of an ideal topological space X,τ,I is semi-I-paracompact.Proof.
LetU be an open semi-I∗-paracompact subset of an ideal topological space X,τ,I. Let V=Vγ|γ∈Γ be any open cover of U by open subsets of the subspace U,τ|U,I|U. Since U is open, V is a cover of U by open subsets of X and so V has a locally finite semi-I-open refinement of W in X,τ,I. Then, W′0=W′∩U|W′∈W is a locally finite semi-I-open refinement of V in U,τ|U,I|U and the result follows.Theorem 19.
LetU be a clopen subspace of an ideal topological space X,τ,I. Then, U is semi-I∗-paracompact if and only if it is semi-I-paracompact.Proof.
It follows from Theorem18.
Conversely, letV=Vγ|γ∈Γ be any open cover of U by open subsets of X,τ,I. Then, V0=U∩Vγ|γ∈Γ is an open cover of the semi-I|U-paracompact subspace U,τ|U,I|U and so V0 has a locally finite semi-I-open refinement W in U,τ|U,I|U. By Lemma 7, W′∈sIOX,τ for every W′∈W. To show that W is locally finite in X,τ,I, let x∈X. If x∈U, then there exists Gx∈τ|U⊆τ containing x such that Gx intersects at most finitely many members of W. Otherwise, X−U is an open set containing x which intersects no member of W. Therefore, W is locally finite in X,τ,I such that U⊆∪W′|W′∈W. Thus, U is semi-I∗-paracompact.Corollary 2.
Every clopen subspace of a semi-I-paracompact space is semi-I-paracompact.



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