On some spaces via topological ideals

Chawalit Boonpok

doi:10.1515/math-2023-0118

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On some spaces via topological ideals

Chawalit Boonpok

Published/Copyright: October 4, 2023

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$Open Mathematics$

From the journal Open Mathematics Volume 21 Issue 1

Abstract

Our main purpose is to introduce and investigate the concepts of some forms of spaces via topological ideals. Some characterizations of some forms of spaces via topological ideals are established. Moreover, several properties of ⋆ -monotonically normal ideal topological spaces are considered.

Keywords: wM(*)-space; sym-wκ-space; *-expandable space; *-monotonically normal space

MSC 2010: 54C08; 54D15; 54D20; 54E18

1 Introduction

The research in point-set topology has been devoted to the study of certain generalization of metrizable spaces. In [1], Morita proved that a T 1 -space X is metrizable if and only if there is a sequence { C i } of locally finite closed coverings of X such that for any point x and for any neighborhood V of x , there is some i for which St ( x , C i ) ⊆ V . The concept of M -spaces was first introduced by Morita [2]. In 1967, Morita [3] introduced the notion of M * -spaces, which contain all M -spaces, and investigated some properties of M * -spaces. In 1970, Ishii [4] introduced the notion of w M -spaces, which is a generalization of M -spaces due to Morita [2], and studied some properties of w M -spaces. Ishii and Shiraki [5] investigated further properties of w M -spaces. Most of which are concerned with metrization of w M -spaces. Ishii [6] studied the metrizability of w M -spaces and proved some metrization theorems for w M -spaces. Monotone normality was introduced by Heath et al. [7]. The concept of monotone normality is a strengthening of normality. Monotone normality was first examined by Borges [8] and later widely investigated in many studies. In 1991, Marín and Romaguera [9] introduced and investigated the notion of pairwise monotonically normal spaces due to Heath et al. [7]. In 1995, Stares [10] provided a characterization of monotone normality with an analogue of the Tietze-Urysohn theorem for monotonically normal spaces as well as an answer to a question due to San-ou concerning the extension of Urysohn functions in monotonically normal spaces. In 2007, Gao et al. [11] showed that a monotonically normal space X is paracompact if and only if for every increasing open cover { U α : α < κ } of X , there is a closed cover { F n α : n < ω , α < κ } of X such that F n α ⊆ U α for n < ω , α < κ , and F n α ⊆ F n β if α ≤ β . In 2017, Sun et al. [12] defined monotone normality in generalized topological spaces and investigated some properties of μ -monotonically normal spaces. In general topology, by introducing the notion of ideal, Kuratowski [13], Vaidyanathaswamy [14,15], and several other authors carried out such analyses. Janković and Hamlett [16] introduced the notion of I -open sets in ideal topological spaces. Abd El-Monsef et al. [17] further investigated I -open sets and I -continuous functions. Later, several authors studied ideal topological spaces giving several convenient definitions. Some authors obtained decompositions of continuity. For instance, Açikgöz et al. [18] introduced and investigated the notions of weakly- I -continuous and weak ⋆ - I -continuous functions in ideal topological spaces. In this study, we introduce and investigate the notions of some forms of spaces via topological ideals. Furthermore, some characterizations of some forms of spaces via topological ideals are investigated. In particular, several properties of ⋆ -monotonically normal ideal topological spaces are discussed.

2 Preliminaries

We begin with some definitions and known results, which will be used throughout this article. In this article, spaces ( X , τ ) and ( Y , σ ) (or simply X and Y ) always mean topological spaces on which no separation axioms are assumed unless explicitly stated. For a subset A of a topological space ( X , τ ) , the closure of A and the interior of A are denoted by Cl ( A ) and Int ( A ) , respectively. An ideal I on a topological space ( X , τ ) is a nonempty collection of subsets of X satisfying the following properties: (1) A ∈ I and B ⊆ A imply B ∈ I and (2) A ∈ I and B ∈ I imply A ∪ B ∈ I . A topological space ( X , τ ) with an ideal I on X is called an ideal topological space and is denoted by ( X , τ , I ) . For an ideal topological space ( X , τ , I ) and a subset A of X , A ⋆ ( I ) is defined as follows: A ⋆ ( I ) = { x ∈ X : U ∩ A ∉ I for every open neighborhood U of x } . In case there is no chance for confusion, A ⋆ ( I ) is simply written as A ⋆ . In [13], A ⋆ is called the local function of A with respect to I and τ and Cl ⋆ ( A ) = A ⋆ ∪ A defines a Kuratowski closure operator for a topology τ ⋆ ( I ) finer than τ , generated by the base B ( I , τ ) = { U − I ′ ∣ U ∈ τ and I ′ ∈ I } . However, B ( I , τ ) is not always a topology [14]. A subset A is said to be ⋆ -closed [16] if A ⋆ ⊆ A . The interior of a subset A in ( X , τ ⋆ ( I ) ) is denoted by Int ⋆ ( A ) .

3 Some forms of spaces via topological ideals

Let U be a cover of an ideal topological space ( X , τ , I ) . For each A ⊆ X , put

St ( A , U ) = ∪ { U ∈ U ∣ A ∩ U ≠ ∅ } ,

and St k + 1 ( A , U ) = St ( St k ( A , U ) , U ) for all k ∈ N . It is easy to check that Cl ⋆ ( St ( A , U ) ) ⊆ St 2 ( A , U ) if U is a ⋆ -open cover of X .

For a sequence { U n } of ⋆ -open coverings of an ideal topological space ( X , τ , I ) , we shall consider the property ( P 1 ) : if { D n } is a sequence of nonempty subsets of X such that D n + 1 ⊆ D n and D n ⊆ St ( x 0 , U n ) for each n ∈ N and for some point x 0 of X , then ∩ n = 1 ∞ Cl ⋆ ( D n ) ≠ ∅ .

An ideal topological space ( X , τ , I ) is said to be a w ℳ ( ⋆ ) -space if there exists a sequence of ⋆ -open coverings of X satisfying the property ( P 2 ) : if { D n } is a sequence of nonempty subsets of X such that D n + 1 ⊆ D n and D n ⊆ St 2 ( x 0 , U n ) for each n ∈ N and for some point x 0 of X , then ∩ n = 1 ∞ Cl ⋆ ( D n ) ≠ ∅ .

Definition 1

A collection C = { C γ ∣ γ ∈ Γ } of subsets of an ideal topological space ( X , τ , I ) is said to be ⋆ -locally finite if for each x ∈ X , there exists a ⋆ -open set U of X containing x and U intersects C γ for at most finitely many γ .

Theorem 2

For an ideal topological space ( X , τ , I ) , the following properties are equivalent:

( X , τ , I ) is a w ℳ ( ⋆ ) -space with a sequence { U n } of ⋆ -open coverings of X satisfying ( P 2 ) .
There exists a sequence { U n } of ⋆ -open coverings of X such that, for any ⋆ -locally finite sequence { A n } of subsets of X , { St ( A n , U n ) ∣ n ∈ N } is ⋆ -locally in X .
There exists a sequence { U n } of ⋆ -open coverings of X such that, for any discrete sequence { x n } of points of X , { St ( x n , U n ) ∣ n ∈ N } is ⋆ -locally finite in X.

Proof

( 1 ) ⇒ ( 2 ) : Let ( X , τ , I ) be a w ℳ ( ⋆ ) -space with a sequence { U n } of ⋆ -open coverings of X satisfying ( P 2 ) . Then we can prove that, for any ⋆ -locally finite sequence { A n } of subsets of X ,

{ St ( A n , U n ) ∣ n ∈ N }

is ⋆ -locally finite in X . Indeed, if not, then for some ⋆ -locally finite sequence { A n } of subsets of X , { St ( A n , U n ) ∣ n ∈ N } is not ⋆ -locally finite in X . Hence, there exists a point x 0 such that any ⋆ -neighborhood of x 0 intersects infinitely many elements of { St ( A n , U n ) ∣ n ∈ N } . Therefore, for each n ∈ N , there exists i ( n ) ∈ N such that St ( x 0 , U n ) ∩ St ( A i ( n ) , U i ( n ) ) ≠ ∅ , and n < i ( n ) . Let y i ( n ) ∈ St ( x 0 , U n ) ∩ St ( A i ( n ) , U i ( n ) ) . Then, the sequence { y i ( n ) } has a cluster point y 0 in X , and hence, we can select a subsequence { y t ( n ) } of { y i ( n ) } such that y t ( n ) ∈ St ( y 0 , U n ) and i ( n ) < t ( n ) . Since y t ( n ) ∈ St ( A t ( n ) , U t ( n ) ) ⊆ St ( A t ( n ) , U n ) , we have A t ( n ) ∩ St 2 ( y 0 , U n ) ≠ ∅ . Let x t ( n ) ∈ A t ( n ) ∩ St 2 ( y 0 , U n ) . Then, the sequence { x t ( n ) } has a cluster point in X by ( P 2 ) , while it has no cluster point in X by ⋆ -local finiteness of { A t ( n ) } . This is a contradiction. Thus, ( 2 ) holds.

( 2 ) ⇒ ( 3 ) : This implication is obvious.

( 3 ) ⇒ ( 1 ) : Let { U n } be a sequence of ⋆ -open coverings of X such that, for any discrete sequence { x n } of points of X , { St ( x n , U n ) ∣ n ∈ N } is ⋆ -locally finite in X . First, we prove that { U n } satisfies ( P 1 ) . To prove this, assume the converse. Then, there exists a discrete sequence { x n } of points of X such that x n ∈ St ( x 0 , U n ) for each n ∈ N and for some point x 0 of X . Since x 0 ∈ St ( x n , U n ) for each n ∈ N , { St ( x n , U n ) ∣ n ∈ N } is not ⋆ -locally finite in X , while it is ⋆ -locally finite in X by our assumption. This is a contradiction. Hence, { U n } satisfies ( P 1 ) . Next, we prove that { U n } satisfies ( P 2 ) . To prove this, assume the converse. Then, there exists a discrete sequence { x n } of points of X such that x n ∈ St 2 ( x 0 , U n ) for each n ∈ N and for some point x 0 of X . Since St ( x n , U n ) ∩ St ( x 0 , U n ) ≠ ∅ , we can select a point y n ∈ St ( x n , U n ) ∩ St ( x 0 , U n ) for each n ∈ N . Then, the sequence { y n } has a cluster point in X by ( P 1 ) , while it has no cluster point in X , because { St ( x n , U n ) ∣ n ∈ N } is ⋆ -locally finite in X . This is a contradiction. Hence, ( 1 ) holds.□

Theorem 3

For an ideal topological space ( X , τ , I ) , the following properties are equivalent:

( X , τ , I ) is a w ℳ ( ⋆ ) -space.
Each point x of X has a sequence { V n ( x ) } of symmetric ⋆ -neighborhoods
( i . e . y ∈ V n ( x ) implies x ∈ V n ( y ) )
satisfying the property ( P 3 ) : if { x n } is a sequence of points of X such that x n ∈ V n 2 ( x 0 ) for each n ∈ N and for some point x 0 of X , then the sequence { x n } has a cluster point in X, where
V n 2 ( x 0 ) = ∪ { V n ( y ) ∣ y ∈ V n ( x 0 ) } .
Each point x of X has a sequence { V n ( x ) } of symmetric ⋆ -neighborhoods such that, for any ⋆ -locally finite sequence { B n } of subsets of X, { V n ( B n ) ∣ n ∈ N } is ⋆ -locally finite in X, where
V n ( B n ) = ∪ { V n ( y ) ∣ y ∈ B n } .
Each point x of X has a sequence { V n ( x ) } of symmetric ⋆ -neighborhoods such that, for any discrete sequence { x n } of points of X, { V n ( x n ) ∣ n ∈ N } is ⋆ -locally finite in X .

Proof

( 1 ) ⇒ ( 2 ) : Let ( X , τ , I ) be a w ℳ ( ⋆ ) -space with a sequence { U n } of ⋆ -open coverings of X satisfying ( P 2 ) , and put V n ( x ) = St ( x , U n ) for each point x of X and for each n ∈ N . Then, { V n ( x ) } is a sequence of symmetric ⋆ -neighborhoods of x and satisfies ( P 3 ) , because V n 2 ( x ) = St 2 ( x , U n ) .

( 2 ) ⇒ ( 3 ) : This implication can be proved similarly to the proof of the implication ( 1 ) ⇒ ( 2 ) in Theorem 2.

( 3 ) ⇒ ( 4 ) : This implication is obvious.

( 4 ) ⇒ ( 1 ) : Suppose that each point x of X has a sequence { V n ( x ) } of symmetric ⋆ -neighborhoods such that, for any discrete sequence { x n } of points of X , { V n ( x ) ∣ n ∈ N } is ⋆ -locally finite in X . Then, it is easily verified that any sequence { x n } of points of X such that x n ∈ V n ( x 0 ) for some point x 0 of X and for each n ∈ N has a cluster point in X . Further, it may be proved by induction for k that any sequence { x n } of points of X such that x n ∈ V n k ( x 0 ) for some point x 0 of X and for each n ∈ N has a cluster point in X . Now, let us put U n = { Int ⋆ ( V n ( x ) ) ∣ x ∈ X } for each n ∈ N . Then, { U n } satisfies ( P 2 ) , because St 2 ( x , U n ) ⊆ V n 4 ( x ) . Hence, ( 1 ) holds.□

Definition 4

A function f : ( X , τ , I ) → ( Y , σ , J ) is said to be:

⋆ -closed [19] if f ( F ) is ⋆ -closed in ( Y , σ , J ) for each ⋆ -closed subset F of ( X , τ , I ) ;
⋆ -continuous if f − 1 ( V ) is ⋆ -open in ( X , τ , I ) for each ⋆ -open subset V of ( Y , σ , J ) .

Definition 5

An ideal topological space ( X , τ , I ) is said to be countably ⋆ -compact if every countable ⋆ -open cover of X has a finite subcover.

Theorem 6

Let f : ( X , τ , I ) → ( Y , σ , J ) be a ⋆ -closed and ⋆ -continuous surjection such that f − 1 ( y ) is countably ⋆ -compact for each y ∈ Y . If ( X , τ , I ) is a w ℳ ( ⋆ ) -space, then ( Y , σ , J ) is also a w ℳ ( ⋆ ) -space.

Proof

Let { U n } be a decreasing sequence of ⋆ -open coverings of X satisfying ( P 2 ) . For each n ∈ N and for each point y of Y , let us put V n ( y ) = Y − f ( X − St ( f − 1 ( y ) , U n ) ) and B n = { V n ( y ) ∣ y ∈ Y } . Then, it is easy to verify that V n ( y ) are ⋆ -open subsets of Y such that y ∈ V n ( y ) , V n + 1 ( y ) ⊆ V n ( y ) , and f − 1 ( V n ( y ) ) ⊆ St ( f − 1 ( y ) , U n ) . We now prove that the sequence { B n } of ⋆ -open coverings of Y satisfies ( P 2 ) . For this purpose, by Theorem 2, it is sufficient to prove that, for every discrete sequence { y n } of points of Y ,

{ St ( y n , B n ) ∣ n ∈ N }

is ⋆ -locally finite in Y . Suppose that this is not valid for some discrete sequence { y n } of points of Y . Then, there exist a point y 0 of Y and a sequence { n ( i ) ∣ i = 1 , 2 , … } of positive integers such that:

V i ( y 0 ) ∩ St ( y n ( i ) , B n ( i ) ) ≠ ∅ ,

i = 1 , 2 , … , and n ( 1 ) < ⋯ < n ( i ) < ⋯ . Let z i ∈ V i ( y 0 ) ∩ St ( y n ( i ) , B n ( i ) ) . Then, from z i ∈ V i ( y 0 ) , it follows that the sequence { z i } has a cluster point in Y . Indeed, let t i ∈ f − 1 ( z i ) . Then, t i ∈ f − 1 ( V i ( y 0 ) ) ⊆ St ( f − 1 ( y 0 ) , U i ) , and hence, it is easily proved that the sequence { t i } has a cluster point in X , because f − 1 ( y 0 ) is countably ⋆ -compact. Thus, the sequence { z i } has a cluster point in Y . On the other hand, from z i ∈ St ( y n ( i ) , B n ( i ) ) , it follows that the sequence { z i } has no cluster point in Y . Indeed, let u i be the points of Y such that y n ( i ) ∈ V n ( i ) ( u i ) and z i ∈ V n ( i ) ( u i ) . Then, since f − 1 ( V n ( i ) ( u i ) ) ⊆ St ( f − 1 ( u i ) , U n ( i ) ) , the sets f − 1 ( y n ( i ) ) and f − 1 ( z n ( i ) ) are contained in St ( f − 1 ( u i ) , U n ( i ) ) . Thus,

f − 1 ( u i ) ∩ St ( f − 1 ( y n ( i ) ) , U n ( i ) ) ≠ ∅ .

Let s i ∈ f − 1 ( u i ) ∩ St ( f − 1 ( y n ( i ) ) , U n ( i ) ) . Since { f − 1 ( y n ) ∣ n ∈ N } is a discrete collection of subsets of a w ℳ ( ⋆ ) -space ( X , τ I ) , { St ( f − 1 ( y n ) , U n ) ∣ n ∈ N } is ⋆ -locally in X by Theorem 2, and hence, the sequence { s i } has no cluster point in X . Accordingly, the sequence { u i } has no cluster point in Y , because f is ⋆ -closed and u i = f ( s i ) . Therefore, by Theorem 2, { St ( f − 1 ( u i ) , U n ( i ) ) ∣ i = 1 , 2 , … } is ⋆ -locally finite in X . This implies that { f − 1 ( z i ) } is also ⋆ -locally finite in X , because f − 1 ( z i ) ⊆ St ( f − 1 ( u i ) , U n ( i ) ) . Hence, the sequence { z i } has no cluster point in Y , which is a contradiction.□

Recall that a topological space ( X , τ ) is a sym-wg-space [20] if there is a symmetric g -function g : ω × X → τ such that if { x , x n } ⊆ g ( n , y n ) for all n ∈ ω , then { x n } has a cluster point in X .

A κ -function on an ideal topological space ( X , τ , I ) is a mapping κ : N × X → τ ⋆ ( I ) such that x ∈ κ ( n , x ) for each n ∈ N . A κ -function κ on an ideal topological space ( X , τ , I ) is called symmetric if for each n ∈ N and x , y ∈ X , y ∈ κ ( n , x ) whenever x ∈ κ ( n , y ) .

Definition 7

An ideal topological space ( X , τ , I ) with a symmetric κ -function is called sym-w κ -space if κ satisfies: for each n ∈ N and x ∈ X , if { x , x n } ⊆ κ ( n , y n ) , then { x n } has a cluster point.

Definition 8

An ideal topological space ( X , τ , I ) is said to be ⋆ -expandable if for every ⋆ -locally finite collection C = { C γ ∣ γ ∈ Γ } of subsets of X , there exists a ⋆ -locally finite collection G = { G γ ∣ γ ∈ Γ } of ⋆ -open subsets of X such that C γ ⊆ G γ for each γ ∈ Γ .

Theorem 9

Every s y m - w κ -space is ⋆ -expandable.

Proof

Let ( X , τ , I ) be a s y m - w κ -space. Then, there is a symmetric κ -function κ : N × X → τ ⋆ ( I ) such that if { x , x n } ⊆ κ ( n , y n ) for each n ∈ N , then { x n } has a cluster point in X . Without loss of generality, we can assume that each κ ( n + 1 , x ) ⊆ κ ( n , x ) . Put U n = { κ ( n , x ) ∣ x ∈ X } for each n ∈ N . Then, { U n } is a sequence of ⋆ -open covers of X . For a sequence { x n } and x in X , if x n ∈ St 2 ( x , U n ) for all n ∈ N , then { x n } has a cluster point in X . There exist a n , b n , c n ∈ X such that x ∈ κ ( n , a n ) , x n ∈ κ ( n , b n ) , and c n ∈ κ ( n , a n ) ∩ κ ( n , b n ) by x n ∈ St 2 ( x , U n ) . Then, { c n } has a cluster point c in X . Thus, there exists a subsequence { c n i } of { c n } such that c n i ∈ κ ( i , c ) for all i ∈ N ; thus, c ∈ κ ( i , c n i ) and b n i ∈ κ ( n i , c n i ) ⊆ κ ( i , c n i ) for all i ∈ N by the symmetry. Hence, { b n i } has a cluster point in X . By a similar method, { x n } has a cluster point in X . For a sequence { x n } and x in X , if x n ∈ St 3 ( x , U n ) for all n ∈ N , then { x n } has a cluster point in X . If not, then { { x n } ∣ n ∈ N } is ⋆ -locally finite in X , thus { St ( x n , U n ) ∣ n ∈ N } is ⋆ -locally finite in X . Otherwise, there exists z ∈ X such that for each n ∈ N , St ( z , U n ) ∩ St ( x i ( n ) , U i ( n ) ) ≠ ∅ for some i ( n ) ≥ n . Then, x i ( n ) ∈ St ( St ( z , U n ) , U i ( n ) ) ⊆ St 2 ( z , U n ) , and { x i ( n ) } has a cluster point in X , which is a contradiction. Now, St 2 ( x , U n ) ∩ St ( x n , U n ) ≠ ∅ by x n ∈ St 3 ( x , U n ) ; thus, take y n ∈ St 2 ( x , U n ) ∩ St ( x n , U n ) for all n ∈ N . Then, { y n } has a cluster point in X , which is a contradiction because { St ( x n , U n ) ∣ n ∈ N } is ⋆ -locally finite in X . If { B n ∣ n ∈ N } is an increasing ⋆ -open cover of X , there exists a ⋆ -locally finite ⋆ -open cover { G n ∣ n ∈ N } of X with each Cl ⋆ ( G n ) ⊆ B n . Put D n = X − St 2 ( X − B n , U n ) for all n ∈ N . Then, { D n ∣ n ∈ N } is a ⋆ -closed cover of X . In fact, if z ∈ X − ∪ n ∈ N D n = ∩ n ∈ N St 2 ( X − B n , U n ) , then St 2 ( z , U n ) ∩ ( X − B n ) ≠ ∅ , and take z n ∈ St 2 ( z , U n ) ∩ ( X − B n ) for all n ∈ N . Let c be a cluster point of { z n } in X . Then, c ∈ ∩ n ∈ N ( X − B n ) , which is a contradiction. Let H n = X − Cl ⋆ ( St ( X − B n , U n ) ) . Then, D n ⊆ H n ⊆ Cl ⋆ ( H n ) ⊆ B n for all n ∈ N . Thus, { H n ∣ n ∈ N } is a ⋆ -open cover of X . Put B n ′ = ∪ i ≤ n H i for each n ∈ N . Then, { B n ′ ∣ n ∈ N } is an increasing ⋆ -open cover of X . By the similar method, there exists a ⋆ -open cover { H n ′ ∣ n ∈ N } of X with each Cl ⋆ ( H n ′ ) ⊆ B n ′ . For each n ∈ N , put G n = H n − ∪ i < n Cl ⋆ ( H n ′ ) , then G n is ⋆ -open, and Cl ⋆ ( G n ) ⊆ B n . { G n ∣ n ∈ N } is a ⋆ -locally finite cover of X . In fact, let x ∈ X , then x ∈ H m ′ for some m ∈ N . Thus, H m ′ ∩ G n = ∅ for each n > m , which implies that { G n ∣ n ∈ N } is ⋆ -locally finite in X . On the other hand, ∪ i < n Cl ⋆ ( H n ′ ) ⊆ ∪ i < n B n ′ = ∪ i < n H i . Thus, H n − ∪ i < n H i ⊆ G n , which implies that { G n ∣ n ∈ N } is a cover of X . ( X , τ , I ) is ⋆ -expandable. Let { C γ ∣ γ ∈ Γ } be a ⋆ -locally finite collection of ⋆ -closed subsets of X . For each n ∈ N , put A n = { x ∈ X ∣ { St 2 ( x , U n ) ∩ C γ ≠ ∅ , γ ∈ Γ } is finite } and B n = Int ⋆ ( A n ) . Then, B n ⊆ B n + 1 . We shall check that { B n ∣ n ∈ N } is a cover of X . In fact, for each z ∈ X , there is n ∈ N such that { St 3 ( z , U n ) ∩ C γ ≠ ∅ ∣ γ ∈ Γ } is finite, then St ( z , U n ) ⊆ A n ; thus, z ∈ B n . There exists a ⋆ -locally finite ⋆ -open cover { G n ∣ n ∈ N } of X with each Cl ⋆ ( G n ) ⊆ B n . For each γ ∈ Γ , n ∈ N , put G γ , n = St ( C γ , U n ) ∩ G n and H γ = ∪ n ∈ N G γ , n . Then, H γ is ⋆ -open in X , and C γ ⊆ H γ . We shall show that { H γ ∣ γ ∈ Γ } is ⋆ -locally finite in X . In fact, for each x ∈ X , denote { n ∈ N ∣ x ∈ Cl ⋆ ( G n ) } = { n i ∣ i ≤ k } for some k ∈ N , and m ( x ) = max { n i ∣ i ≤ k } . Put V = St ( x , U m ( x ) ) − ∪ { Cl ⋆ ( G n ) ∣ x ∉ Cl ⋆ ( G n ) , n ∈ N } . Then, V is a ⋆ -open neighborhood of x in X . For each i ≤ k , { St ( x , U m ( x ) ) ∩ G γ , n i ≠ ∅ ∣ γ ∈ Γ } ⊆ { St 2 ( x , U n i ) ∩ C γ ≠ ∅ ∣ γ ∈ Γ } is finite by x ∈ B n i ; thus, V only meets with finitely many elements of { H γ ∣ γ ∈ Γ } . Thus, ( X , τ , I ) is ⋆ -expandable.□

4 On ⋆ -monotonically normal ideal topological spaces

In this section, we introduce the notion of ⋆ -monotonically normal ideal topological spaces. Furthermore, some characterizations of ⋆ -monotonically normal ideal topological spaces are investigated.

Definition 10

An ideal topological space ( X , τ , I ) is said to be ⋆ -monotonically normal if there is a function H , which assigns to each ordered pair ( A , B ) of disjoint ⋆ -closed subsets of X a ⋆ -open set H ( A , B ) such that:

A ⊆ H ( A , B ) ⊆ Cl ⋆ ( H ( A , B ) ) ⊆ X − B ;
H ( A , B ) ⊆ H ( A ′ , B ′ ) , whenever A ⊆ A ′ and B ′ ⊆ B .

The function H is called a ⋆ -monotone normality operator for X .

Lemma 11

If ( X , τ , I ) is a ⋆ -monotonically normal space, then there is a ⋆ -monotone normality operator H satisfying H ( A , B ) ∩ H ( B , A ) = ∅ for each pair ( A , B ) of disjoint ⋆ -closed sets.

Proof

Let H ′ be any ⋆ -monotone normality operator for X , and define H ( A , B ) = H ′ ( A , B ) − Cl ⋆ ( H ′ ( B , A ) ) . Then, H is a ⋆ -monotone normality operator for X having H ( A , B ) ∩ H ( B , A ) = ∅ .□

Definition 12

An ideal topological space ( X , τ , I ) is called ⋆ - T 1 if for any pair of distinct points x and y of X , there exist a ⋆ -open set U of X containing x but not y and a ⋆ -open set V of X containing y but not x .

Lemma 13

An ideal topological space ( X , τ , I ) is ⋆ - T 1 if and only if the singletons of X are ⋆ -closed.

Theorem 14

Let ( X , τ , I ) be a ⋆ - T 1 ideal topological space. The following properties are equivalent:

( X , τ , I ) is ⋆ -monotonically normal.
To each pair ( S , T ) of subsets of X, with S ∩ Cl ⋆ ( T ) = ∅ = Cl ⋆ ( S ) ∩ T , one can assign a ⋆ -open subset H 1 ( S , T ) of X such that:
1. S ⊆ H 1 ( S , T ) ⊆ Cl ⋆ ( H 1 ( S , T ) ) ⊆ X − T ;
2. H 1 ( S , T ) ⊆ H 1 ( S ′ , T ′ ) , whenever S ⊆ S ′ and T ′ ⊆ T .
To each ordered pair ( F , U ) of subsets of X with F ⋆ -closed, U ⋆ -open, and F ⊆ U , we can assign a ⋆ -open set H 2 ( F , U ) such that:
1. F ⊆ H 2 ( F , U ) ⊆ Cl ⋆ ( H 2 ( F , U ) ) ⊆ U ;
2. if H is ⋆ -closed, V is ⋆ -open, H ⊆ V , F ⊆ H , and U ⊆ V , then H 2 ( F , U ) ⊆ H 2 ( H , V ) .
For each pair ( x , U ) , where U is a ⋆ -open set containing x, there is a ⋆ -open set H 3 ( x , U ) such that:
1. x ∈ H 3 ( x , U ) ⊆ U ;
2. if x ∈ U ⊆ V , then H 3 ( x , U ) ⊆ H 3 ( x , V ) for every ⋆ -open set V ;
3. if H 3 ( x , U ) ∩ H 3 ( y , V ) ≠ ∅ , then either x ∈ V or y ∈ U .
There is a function L that assigns to each ordered pair ( p , C ) , with C ⋆ -closed and p ∈ X − C , a ⋆ -open set L ( p , C ) satisfying
1. p ∈ L ( p , C ) ⊆ X − C ;
2. if D is ⋆ -closed and p ∉ C ⊇ D , then L ( p , C ) ⊆ L ( p , D ) ;
3. if p ≠ q are the points of X, then L ( p , { q } ) ∩ L ( q , { p } ) = ∅ .

Proof

( 1 ) ⇒ ( 3 ) : Let H be any ⋆ -monotone normality operator for X . To each ordered pair ( F , U ) of subsets of X with F ⋆ -closed, U ⋆ -open, and F ⊆ U , let us assign H 2 ( F , U ) = H ( F , X − U ) . It is easy to check that conditions (i) and (ii) of ( 3 ) are satisfied.

( 3 ) ⇒ ( 4 ) : For each pair ( x , U ) , where U is a ⋆ -open set containing x , since ( X , τ , I ) is a ⋆ - T 1 space, { x } is ⋆ -closed. Let H 3 ( x , U ) = H 2 ( { x } , U ) , where the operator H 2 is defined as in ( 3 ) . By the definition of H 2 in ( 3 ) , we have x ∈ H 3 ( x , U ) ⊆ U , and if x ∈ U ⊆ V , then H 3 ( x , U ) ⊆ H 3 ( x , V ) . Suppose that H 3 ( x , U ) ∩ H 3 ( y , V ) ≠ ∅ . Then, x ∉ V and y ∉ U . Therefore, { x } ⊆ X − V and { y } ⊆ X − U . Thus, putting H ( A , B ) = H 2 ( A , X − B ) , by Lemma 11,

H 3 ( x , U ) ∩ H 3 ( y , V ) = H 2 ( { x } , U ) ∩ H 2 ( { y } , V ) = H ( { x } , X − U ) ∩ H ( { y } , X − V ) ⊆ H ( X − V , { y } ) ∩ H ( { y } , X − V ) = ∅ ,

which is a contradiction. This proves that H 3 ( x , U ) ∩ H 3 ( y , V ) ≠ ∅ implies x ∈ V or y ∈ U .

( 4 ) ⇒ ( 5 ) : For each ordered pair ( p , C ) , with C ⋆ -closed and p ∈ X − C , obviously, X − C is ⋆ -open. Let L ( p , C ) = H 3 ( p , X − C ) , where H 3 satisfies Condition ( 4 ) . It is clear that Conditions (i) and (ii) of ( 5 ) hold by the Conditions (i) and (ii) of ( 4 ) . Now, we prove that L ( p , { q } ) ∩ L ( q , { p } ) = ∅ . Suppose that L ( p , { q } ) ∩ L ( q , { p } ) ≠ ∅ , that is, H 3 ( p , X − { q } ) ∩ H 3 ( q , X − { p } ) ≠ ∅ , which implies p ∈ X − { p } or q ∈ X − { q } by (iii) of Condition ( 4 ) , which is a contradiction. Thus, L ( p , { q } ) ∩ L ( q , { p } ) = ∅ .

( 5 ) ⇒ ( 2 ) : For any pair ( S , T ) of subsets of X , with S ∩ Cl ⋆ ( T ) = Cl ⋆ ( S ) ∩ T = ∅ , let

H 1 ( S , T ) = ∪ { L ( p , Cl ⋆ ( T ) ) ∣ p ∈ S } .

Obviously, S ⊆ H 1 ( S , T ) . For each q ∈ T , the set L ( q , Cl ⋆ ( S ) ) is a ⋆ -open set containing q . It follows from (ii) of ( 5 ) that L ( p , Cl ⋆ ( T ) ) ⊆ L ( p , { q } ) and L ( q , Cl ⋆ ( S ) ) ⊆ L ( q , { p } ) if p ∈ S ⊆ Cl ⋆ ( S ) and q ∈ T ⊆ Cl ⋆ ( T ) . By (iii) of ( 5 ) , we have for q ∈ T that:

L ( q , Cl ⋆ ( S ) ) ∩ H 1 ( S , T ) = ∪ p ∈ S ( L ( q , Cl ⋆ ( S ) ) ∩ L ( p , Cl ⋆ ( T ) ) ) ⊆ ∪ p ∈ S ( L ( p , { q } ) ∩ L ( q , { p } ) ) = ∅ .

Thus, q ∉ Cl ⋆ ( H 1 ( S , T ) ) , and hence, Cl ⋆ ( H 1 ( S , T ) ) ⊆ X − T . (ii) of ( 2 ) follows directly from (ii) of ( 5 ) .

( 2 ) ⇒ ( 1 ) : For any pair ( A , B ) of disjoint ⋆ -closed sets of X , obviously, we have

A ∩ Cl ⋆ ( B ) = Cl ⋆ ( A ) ∩ B = ∅ .

Thus, ( 2 ) clearly implies ( 1 ) .□

Definition 15

An ideal topological space ( X , τ , I ) is said to have the ℬ ⋆ -property if every increasing ⋆ -open cover { U α ∣ α < λ } of X has an increasing ⋆ -open cover { V α ∣ α < λ } of X such that Cl ⋆ ( V α ) ⊆ U α for each α < λ , where by { U α ∣ α < λ } being increasing we mean U α ⊆ U β wherever α < β .

Theorem 16

Let ( X , τ , I ) be a ⋆ -monotonically normal space. Then, the following properties are equivalent:

( X , τ , I ) has the ℬ ⋆ -property.
Every increasing ⋆ -open cover { U α ∣ α < λ } of X has a ⋆ -closed cover { C n α ∣ n < ω , α < λ } of X such that C n α ⊆ U α for n < ω , α < λ , and C n α ⊆ C n β if α ≤ β .

Proof

( 1 ) ⇒ ( 2 ) : Let { U α ∣ α < λ } be an increasing ⋆ -open cover of X . By ( 1 ) , there exists an increasing ⋆ -open cover { V α ∣ α < λ } of X with Cl ⋆ ( V α ) ⊆ U α for each α < λ . Put C n α = Cl ⋆ ( V α ) for n < ω , α < λ , then the ⋆ -closed cover { C n α ∣ n < ω , α < λ } of X is the desired one.

( 2 ) ⇒ ( 1 ) : Let { U α ∣ α < λ } be an increasing ⋆ -open cover of X . By ( 2 ) , there exists a ⋆ -closed cover

{ C n α ∣ n < ω , α < λ }

of X such that C n α ⊆ U α for n < ω , α < λ , and C n α ⊆ C n β if α ≤ β . Let H 0 be a ⋆ -monotone normality operator for the space X . Since C n α ⊆ U α , for the disjoint ⋆ -closed sets C n α and X − U α , we have

C n α ⊆ H 0 ( C n α , X − U α ) ⊆ Cl ⋆ ( H 0 ( C n α , X − U α ) ) ⊆ U α ,

n < ω . Since the ⋆ -open cover { U α ∣ α < λ } of X is increasing and C n α ⊆ C n β for each α ≤ β , it holds that H 0 ( C n α , X − U α ) ⊆ H 0 ( C n β , X − U β ) for each α ≤ β . For n < ω and α < λ , put G n α = H 0 ( C n α , X − U α ) . Thus, C n α ⊆ G n α ⊆ Cl ⋆ ( G n α ) ⊆ U α for n < ω , α < λ , and G n α ⊆ G n β for each α ≤ β . For n < ω and α < λ , let K n α = G n α − ∪ { G i β ∣ i < n , β < λ } and L α = Cl ⋆ ( ∪ n < ω K n α ) . Then, K n α ⊆ K n β , α ≤ β , and n < ω , so L α ⊆ L β for each α ≤ β . We claim that { L α ∣ α < λ } is a ⋆ -closed cover of X . In fact, it can be checked that ∪ α < λ L α ⊇ ∪ α < λ ∪ n < ω K n α = ∪ n < ω ∪ α < λ K n α = ∪ n < ω ∪ α < λ G n α ⊇ ∪ n < ω ∪ α < λ C n α = X . Now, we shall show that for each α < λ , L α ⊆ U α . Let x ∉ U α . Since { G n α ∣ n < ω , α < λ } is a ⋆ -open cover of X and Cl ⋆ ( G n α ) ⊆ U α for each n < ω , there is a ⋆ -open set G n ′ β ′ such that x ∈ G n ′ β ′ and β ′ ≠ α . If m > n ′ , then K m α ∩ G n ′ β ′ = ∅ . If m ≤ n ′ , since ∪ n ≤ n ′ Cl ⋆ ( G n α ) ⊆ U α , we have x ∈ X − ∪ n ≤ n ′ Cl ⋆ ( G n α ) . Consequently, we obtain ( X − ∪ n ≤ n ′ Cl ⋆ ( G n α ) ) ∩ K m α = ∅ . Put W = G n ′ β ′ ∩ ( X − ∪ n ≤ n ′ Cl ⋆ ( G n α ) ) . Then, for each m < ω , W ∩ K m α = ∅ and so W ∩ ( ∪ n < ω K n α ) = ∅ . Hence, x ∉ L α . Thus, for disjoint ⋆ -closed sets L α and X − U α , we have L α ⊆ H 0 ( L α , X − U α ) ⊆ Cl ⋆ ( H 0 ( L α , X − U α ) ) ⊆ U α . Put D α = H 0 ( L α , X − U α ) , α < λ . Then, { D α ∣ α < λ } is an increasing ⋆ -open cover of X , which refines the increasing ⋆ -open cover { U α ∣ α < λ } of X with Cl ⋆ ( D α ) ⊆ U α for each α < λ . Thus, ( X , τ , I ) has the ℬ ⋆ -property, and thus, ( 2 ) holds.□

Recall that if f : ( X , τ ) → ( Y , σ ) is a continuous closed surjection and ( X , τ ) is monotonically normal, then ( Y , σ ) is monotonically normal [7].

Theorem 17

Let f : ( X , τ , I ) → ( Y , σ , J ) be a ⋆ -closed and ⋆ -continuous surjection. If ( X , τ , I ) is ⋆ -monotonically normal, then ( Y , σ , J ) is ⋆ -monotonically normal.

Proof

Let H be a ⋆ -monotone normality operator for X . For disjoint ⋆ -closed subsets F 1 and F 2 of Y , since f is ⋆ -continuous, f − 1 ( F 1 ) and f − 1 ( F 2 ) are disjoint ⋆ -closed subsets of X . Put

H ′ ( F 1 , F 2 ) = Y − f ( X − H ( f − 1 ( F 1 ) , f − 1 ( F 2 ) ) ) .

Then, H ′ is a ⋆ -monotone normality for Y , and hence, ( Y , σ , J ) is a ⋆ -monotonically normal space.□

Acknowledgement

The author is very grateful to the referees for the careful reading of this manuscript and offering valuable comments for this manuscript.

Funding information: This research project was financially supported by Thailand Science Research and Innovation (TSRI).
Author contributions: The author has accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The author states no conflict of interest.
Data availability statement: No data were used to support this study.

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Received: 2023-02-16

Revised: 2023-08-21

Accepted: 2023-08-22

Published Online: 2023-10-04

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/math-2023-0118

Keywords for this article

wM(*)-space; sym-wκ-space; *-expandable space; *-monotonically normal space

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