Properties of locally semi-compact Ir-topological groups

ZhongLi Wang; Wen Chean Teh

doi:10.1515/math-2023-0144

Open Access Published by De Gruyter Open Access November 20, 2023

Properties of locally semi-compact Ir-topological groups

ZhongLi Wang and Wen Chean Teh

From the journal Open Mathematics

https://doi.org/10.1515/math-2023-0144

Abstract

This study investigates some topological properties of locally semi-compact Ir-topological groups and establishes the relationship between Ir-topological groups and semi-compact spaces. The proved theorems generalize the corresponding results of Ir-topological group. Finally, we define a quotient topology on the Ir-topological group and study some topological properties of the space.

Keywords: semi-open sets; locally semi-compact spaces; Ir-topological groups

MSC 2010: 22B05; 22B10; 22D05; 54A05

1 Introduction

Recently, topological groups have garnered significant attention from topologists due to their applications in graph algebras and the study of hyperbolic groups [1–6]. Many fundamental concepts and constructions related to topological groups have been introduced. One of the generic questions in topological algebra is how the relationship between topological properties depends on the underlying algebraic structure [7]. Locally compact groups are essential because many examples of groups that arise throughout mathematics are locally compact [8–11]. The rules that describe the relationship between a locally compact group and an algebraic operation are almost always continuous. It is natural to explore the properties of topological groups by relaxing the continuity conditions. Consequently, Levine [12] introduced the concepts of semi-open sets and semi-continuity within general topological spaces. These concepts have now become research topics for topologists worldwide, including the study of semi-separation axioms and binary topological spaces [13–15].

One of the critical applications of semi-open sets is compactness. In 1984, the definition of locally semi-compact spaces [16] was introduced. Furthermore, many scholars use semi-open sets to study topological groups, and the study of the topological group is wide open. In 2014, Bosan et al. [17] studied the class of s-topological groups and a wider class of S-topological groups defined using semi-open sets and semi-continuity. In 2015, two types of topological groups, which are called irresolute-topological and Ir-topological [18], were introduced and studied.

These groups form a generalization of topological groups, and some preliminary results and applications of these groups are presented. However, the relationships between these topological groups and other topological spaces have yet to be obtained. One of the main operations on topological groups is taking quotient groups. The quotient spaces of some topological groups have yet to be proposed either.

To solve these problems mentioned above, the primary purpose of this study is to establish relationships between Ir-topological groups and semi-compact spaces and define a quotient topology on the Ir-topological group. We will introduce some basic properties of locally semi-compact spaces and Ir-topological groups in Section 2. In Section 3, by introducing the concept of countable semi-stars, the connection between Ir-topological groups and semi-compact spaces is constructed, and the concept of quotient spaces is also introduced. Also, we generalize a series of results on Ir-topological groups.

Throughout this study, X and Y are always topological spaces on which no separation axioms are assumed. The set of positive integers is denoted as N and the real line is denoted as R . The family of all semi-open sets in X is denoted by SO ( X ) . The interior and semi-interior of A in X are denoted as A 0 and sop ( A ) . The closure and semi-closure of A in X are denoted as A ¯ and scl A . In this study, G denotes a group endowed with a topology. We do not require the group operations to be continuous but to satisfy some weaker forms of continuity. If G is a group, then e denotes its identity element and τ denotes its topology. For definitions not defined here, we refer the reader to [7].

2 Preliminaries

Ir-topological groups are one of the essential classes of topological groups, and many results have been obtained. We study in this section the most elementary general properties of locally semi-compact spaces and Ir-topological groups and obtain some new properties.

We need to recall some basic notations. A subset A of a topological space X is said to be semi-open [12] if there exists an open set B in X such that B ⊂ A ⊂ B ¯ . Obviously, any open set is semi-open, but the converse need not be true. The complement of a semi-open set is said to be semi-closed. A collection B of subsets of a topological space X is said to be s-locally finite [19] if for each x ∈ X , there exists a semi-open set U in X containing x and U intersects at most finitely many members of B . Clearly, if a collection is locally finite, it is s-locally finite, but the converse need not be true.

A collection C of subsets of a topological space X is a net [20] if for each point x ∈ X and any open set V containing x , there exists a set C ∈ C such that x ∈ C ⊂ V . A topological space X is said to be s-regular [21] if for any closed set B and x ∉ B , there exist disjoint semi-open sets U and V such that x ∈ U and B ⊂ V .

A topological space X is said to be semi-compact [22] if every cover of X by semi-open sets has a finite subcover. A topological space X is said to be locally semi-compact [16] if every point of X has an open semi-compact neighborhood. Obviously, each semi-compact space is a locally semi-compact space. But the converse need not be true. For example, let X be an infinite discrete topological space. Thus, X is not semi-compact. Since the singletons can serve as semi-compact neighborhoods, it follows that X is locally semi-compact.

Proposition 2.1

Suppose X is a locally semi-compact s-regular space. Then, the collection of semi-compact semi-closed sets in X is a net.

Proof

Suppose x in X and A is an open set containing x . Since X is locally semi-compact, it follows that there is an open neighborhood B which is semi-compact. Then, A ∩ B is a neighborhood of x . Let C = X − A ∩ B . Then, x is not in C , and C is closed. Since X is s-regular, it follows that there exist disjoint semi-open sets E and F such that x in E and C ⊂ F . Let D = X − F . Then, D is semi-closed and X − F ⊂ X − C . Thus, D ⊂ A ∩ B and x ∈ D ⊂ A .

It remains to show that D is semi-compact. Suppose { H α : α ∈ I } is a semi-open cover of D . For each H α , there exists an open set L α ⊂ D such that L α ⊂ H α ⊂ L α ¯ . Then, there exists an open set M α in X such that L α = M α ∩ D . Thus, H α ¯ ⊂ L α ¯ ⊂ M α ¯ . Let O α = M α ∪ H α . Then, O α ∩ D = H α . Since M α ⊂ O α ⊂ O α ¯ and O α ¯ = M α ∪ H α ¯ = M α ¯ , it follows that O α is semi-open in B . Then, { O α : α ∈ I } ∪ { B − D } is a semi-open cover of B . There exists a semi-open cover { O 1 , O 2 , ⋯ , O r } ∪ { B − D } in B . Thus, there exists a semi-open cover { H 1 , H 2 , ⋯ , H r } of D , and D is semi-compact.□

Definition 2.2

[23] A subset A of a topological space X is said to be pre-open if A ⊂ A ¯ 0 .

Obviously, each open set is pre-open, but the converse need not be true. Also, we note that the notions of pre-open and semi-open are independent of each other. For example, let X = { x 1 , x 2 , x 3 , x 4 } and τ = { X , ∅ , { x 1 , x 4 } , { x 3 } , { x 1 , x 3 , x 4 } } . Then, ( X , τ ) is a topological space, and { x 1 , x 2 , x 3 } is a pre-open set, but it is not semi-open. Since { x 3 } ⊂ { x 2 , x 3 } ⊂ { x 3 } ¯ and { x 2 , x 3 } ¯ 0 = { x 3 } , it follows that { x 2 , x 3 } is semi-open but not pre-open. From the definition of pre-open, we have the following remark.

Remark 2.3

If A is pre-open in topological space X , then there exists an open set B in X such that A ⊂ B ⊂ A ¯ .

Definition 2.4

[24] A mapping f : X → Y is said to be irresolute (respectively, pre-semi-open) if f − 1 ( V ) (respectively, f ( V ) ) is semi-open in X (respectively, Y ) for each semi-open set V in Y (respectively, X ).

Definition 2.5

A collection C of subsets of a topological space X is a semi-net if for each point x ∈ X and any semi-open set V containing x , there exists a set C ∈ C such that x ∈ C ⊂ V .

Obviously, each semi-net in a topological space is a net, but the following example shows that the converse need not be true.

Example 2.6

Suppose X = { x 1 , x 2 , x 3 } and B 1 = { ∅ , X , { x 1 } , { x 2 } , { x 1 , x 2 } } is a topology of X . Thus, B 2 = { ∅ , X , { x 1 } , { x 2 } , { x 1 , x 2 } , { x 1 , x 3 } , { x 2 , x 3 } } is the family of all semi-open sets of X . Suppose B 3 = { X , { x 1 } , { x 2 } } . Then, B 3 is a net of X . Note that { x 1 , x 3 } is a semi-open set and x 3 ∈ { x 1 , x 1 } . For each set, A in B 3 does not satisfy the condition that x 3 ∈ A ⊂ { x 1 , x 3 } . Therefore, B 3 is not a semi-net of X .

Proposition 2.7

Suppose X is a countably semi-compact space and f : X → Y is an irresolute injection. If X has a countable semi-net, then Y is locally semi-compact.

Proof

Assume that B is a countable semi-net of X . Suppose A is a semi-open cover of X . For each point x in A and A ∈ A , there exists a set B in B such that x in B and B ⊂ A . Then, there exists a family B A ⊂ B such that A = ∪ B ∈ B A B . Let ∪ A ∈ A B A = B 0 . Then, B 0 is a countable cover of X . Let B 0 = { B n : n ∈ N } . If B n ∈ B 0 , and then there exists a set A n ∈ A such that B n ⊂ A n . Let

A 1 = { A n : A n ∈ A , B n ∈ B 0 , B n ⊂ A n } .

Then, A 1 ⊂ A is a countable cover of X , and X is semi-Lindelöf. Since X is countably semi-compact, it follows that X is semi-compact. Suppose C = { C α : α ∈ I } is a semi-open cover of Y . Hence, { f − 1 ( C α ) : α ∈ I } is a semi-open cover of X . Thus, there exists a finite cover { f − 1 ( C α 1 ) , f − 1 ( C α 2 ) , ⋯ , f − 1 ( C α r ) } . Then, { C α 1 , C α 2 , ⋯ , C α r } is a finite cover of Y . Thus, Y is semi-compact, and Y is locally semi-compact.□

Definition 2.8

[18] A topologized group is said to be an Ir-topological group if both the multiplication mapping and the inverse mapping are irresolute.

Definition 2.9

[24] A mapping f : X → Y is said to be a semi-homeomorphism if f is bijective, irresolute, and pre-semi-open.

Proposition 2.10

Suppose G is an Ir-topological group and each a ∈ X . Then, the right multiplication r a is a semi-homeomorphism.

Proof

Suppose x and y are in G with x ≠ y . Assume that r a ( x ) = r a ( y ) . Then, x a = y a and x = y . It is contradictory, and r a is an injection. For each b in G , there exists a point c = b a − 1 in G such that r a ( c ) = b . Thus, r a is a surjection and r a is bijection.

Define a mapping f 1 : G → G × G , x → ( x , a ) . Let f : G × G → G be the multiplication mapping. Thus, r a ( x ) = f ( f 1 ( x ) ) for each x in G . For each x in G and each semi-open neighborhood A of f 1 ( x ) , there exist semi-open sets B and C such that f 1 ( x ) ∈ B × C ⊂ A . Then, x in B and a in C . Thus, f 1 ( B ) ⊂ A and f 1 is irresolute. Since G is an Ir-topological group, it follows that f is irresolute. Hence, r a is irresolute. Since r a − 1 = r a − 1 , it follows that r a is pre-semi-open. Therefore, r a is a semi-homeomorphism.□

According to Proposition 2.10, we obtain the following remarks directly.

Remark 2.11

Suppose G is an Ir-topological group and each a ∈ X . Then, the left multiplication l a is a semi-homeomorphism.

Remark 2.12

Suppose G is an Ir-topological group and A is semi-open in G . Then, A x and x A are semi-open for each x in G .

3 Properties of Ir-topological group

This section contains theorems on topological properties of locally semi-compact Ir-topological groups. We will use some mappings to investigate the relationships between Ir-topological groups and other topological spaces. Also, we obtained some properties of the Ir-topological group.

It is well known that the intersection of two semi-open sets need not be semi-open. Thus, every family of semi-open sets in a topological space need not be a topology. Example 3.1 [18] shows that even if the family of semi-open sets is a topology on G , the Ir-topological group need not be a topological group.

Example 3.1

The set G = { 1 , 3 , 5 , 7 } is a group under multiplication ⊙ 8 , the multiplication modulo 8. Let τ = { ∅ , G , { 1 } , { 1 , 3 , 5 } } be the topology on G . Then, SO ( G ) = { ∅ , G , { 1 } , { 1 , 3 , 5 } , { 1 , 3 } , { 1 , 5 } , { 1 , 7 } , { 1 , 3 , 7 } , { 1 , 5 , 7 } } .

Thus, the family SO ( G ) is a topology on G , and different from τ . Obviously, each semi-open set in G is pre-open.

For any V ∈ SO ( G ) , the preimage f − 1 ( V ) contains the open set { ( 1 , 1 ) } ⊂ G × G , which is dense in G × G , that is, for each V ∈ SO ( G ) , we have { ( 1 , 1 ) } ⊂ f − 1 ( V ) ⊂ { ( 1 , 1 ) } ¯ . It means that f − 1 ( V ) is semi-open in G × G . Hence, f is irresolute. On the other hand, for each V ∈ SO ( G ) , it holds g − 1 ( V ) = V , which means that g is an irresolute mapping. So, ( G , ⊙ , τ ) is an Ir-topological group which is not a topological group ( f is not continuous, for instance, at ( 3 , 3 ) ∈ G × G ).

According to Example 3.1, we obtain the following proposition directly.

Proposition 3.2

G is a locally semi-compact Ir-topological group, and each semi-open set in G is pre-open, but G does not need to be a topological group.

In order to prove the following result, we need to define some new definitions. A topological space is said to be σ -compact [20] if it is the union of countably many compact subspaces. A topological space is said to be σ semi-compact if it is the union of countably many semi-compact subspaces. Obviously, each σ semi-compact space is σ -compact. The following example shows that each σ -compact space need not be σ semi-compact.

Example 3.3

Let X = R have the standard topology, and Y n = [ − 1 , 1 ] have the subspace topology τ for each n ∈ N . Then, ∪ n ∈ N Y n is σ -compact. Suppose A n = [ − 1 , 0 ] ∪ { ( 1 ⁄ ( n + 1 ) , 1 ⁄ n ) : n ∈ N } . Then, A n is a semi-open cover of Y n and does not have a finite subcover. Hence, ∪ n ∈ N Y n is not σ semi-compact.

Obviously, semi-compact space is σ semi-compact, but the converse need not be true. For example, let X = R have the standard topology. Then, X is σ semi-compact, but not semi-compact. A space X is locally σ semi-compact if for every point x of X , there exists an open neighborhood V such that V is σ semi-compact. A collection is said to be star-finite [20] if every member of the collection intersects only finitely many members. A space X is said to have the countable semi-star property if for each countable star-finite semi-open cover A of X , there exists a finite set B in X such that S t ( B , A ) = ∪ { A : A ∈ A , A ∩ B ≠ ∅ } = X .

Theorem 3.4

Suppose G is a locally σ semi-compact Ir-topological group and each semi-open set is pre-open. If G has the countable semi-star property, then G is semi-compact.

Proof

Since G is a locally σ semi-compact Ir-topological group, it follows that there exists an open neighborhood B of the identity e in G such that B is σ semi-compact. Let B = ∪ n ∈ N B n , and each B n is semi-compact. Then, B is a subgroup of G , and B is semi-Lindelöf. For each a in G − B , according to Proposition 2.10, the set r a ( B ) = B a is semi-open in G . Thus, G − B = ∪ a ∉ B B a is semi-open and B is semi-closed. Since G is the disjoint union of the right cosets of B , it follows that G is semi-homeomorphism to B and G is semi-Lindelöf.

Suppose C ⊂ G is a semi-closed set and e is not in C . Then, D = G − C is an open neighborhood of e . Since the multiplication mapping f : G × G → G , ( x , y ) → x y is irresolute, it follows that f − 1 ( D ) is a semi-open set containing ( e , e ) . Thus, there exist semi-open sets E and F such that e in E , e in F , and E × F ⊂ f − 1 ( D ) . Then, there exist open sets E 1 such that E 1 ⊂ E ⊂ E 1 ¯ and F ⊂ F ¯ 0 ⊂ F ¯ . Hence, E ∩ F ⊂ E 1 ¯ ∩ F ¯ 0 . Suppose x 0 in E 1 ¯ ∩ F ¯ 0 and U x 0 is a neighborhood of x 0 . Thus, U x 0 ∩ F ¯ 0 is a neighborhood of x 0 . Then, U x 0 ∩ F ¯ 0 ∩ E 1 ≠ ∅ and x 0 ∈ E 1 ∩ F ¯ 0 ¯ . Hence, E ∩ F ⊂ E 1 ∩ F ¯ 0 ¯ . Suppose y 0 in E 1 ∩ F ¯ 0 . Then, y 0 is in E , and y 0 is not in X − F ¯ 0 . Thus, there exists a neighborhood V y 0 of y 0 such that V y 0 ∩ ( X − F ¯ 0 ) = ∅ . Assume that y 0 is not in F . Then, y 0 ∈ F ¯ − F = F r { F } . Hence, y 0 in F r { F ¯ 0 } and V y 0 ∩ ( X − F ¯ 0 ) ≠ ∅ . It is contradictory. Then, y 0 ∈ F and E 1 ∩ F ¯ 0 ⊂ E ∩ F . Thus,

E 1 ∩ F ¯ 0 ⊂ E ∩ F ⊂ E 1 ∩ F ¯ 0 ¯ .

Hence, E ∩ F is semi-open.

Let H = E ∩ F . Then, H is a semi-open set containing e and H × H ⊂ f − 1 ( D ) . Then, H 2 ⊂ D . Suppose b in scl H . Since e in H − 1 , it follows that b in b H − 1 and b H − 1 is semi-open. Hence, b H − 1 ∩ H ≠ ∅ . Suppose c in b H − 1 ∩ H . Then, c = b h − 1 , and b = c h ∈ H 2 . Thus, scl H ⊂ H 2 ⊂ D . Suppose L = G − scl H . Then, L is semi-open and C ⊂ L . Since H ⊂ scl H and e in H , it follows that H ∩ L = ∅ .

Suppose O and P are disjoint semi-closed sets, and x in O . Then, x in X − P and X − P is semi-open. Then, there exists a semi-open set Q x such that x in Q x and Q x ⊂ scl Q x ⊂ X − P and scl Q x ∩ P = ∅ . Hence, Q = { Q x : x ∈ O } is a semi-open cover of O . Then, there exists a semi-open set R y such that scl R y ∩ O = ∅ for each y in P . Let R = { R y : y ∈ P } . Then, R is a semi-open cover of P . Thus, Q ∪ R ∪ { X − P ∪ O } is a semi-open cover of G . Then, there exists a countable subcover of G . Thus, there exists a countable semi-open cover { Q n : n ∈ N } of O and a countable semi-open cover { R n : n ∈ N } of P . Let

S n = Q n − ∪ { scl R k : k ≤ n } ,

T n = R n − ∪ { scl Q k : k ≤ n } .

Then, S n and T n are semi-open sets. Hence, S n ∩ T m = ∅ , n in N and m in N . Let S = ∪ n ∈ N S n , T = ∪ n ∈ N T n . Then, O ⊂ S , P ⊂ T , and S ∩ T = ∅ . Thus, G is semi-normal.

Now, we will show that G is semi-compact. Suppose U is a semi-open cover of G . Then, there exists a countable cover M = { M n : n ∈ N } and M ⊂ U . Suppose x in M n . Then, there exists a semi-open set N x such that x in N x and N x ⊂ scl N x ⊂ M n . Thus, { N x : x ∈ G } is a semi-open cover of G . Then, there exists a countable subcover N = { N x n : x ∈ G } of G . For each N x n in N , pick M n in M such that N x n ⊂ M n . Thus, scl N x n ⊂ M n and { scl N x n : n ∈ N } is a semi-closed cover of G . Since G is semi-normal, it follows that there exists a semi-open set U ( n , 1 ) such that scl N x n ⊂ U ( n , 1 ) ⊂ scl U ( n , 1 ) ⊂ M n . Arguing by induction, there exists a family { U ( n , k ) : k ∈ N } such that

scl N x n ⊂ U ( n , 1 ) ⊂ scl U ( n , 1 ) ⊂ U ( n , 2 ) ⊂ ⋯ ⊂ U ( n , k ) ⊂ scl U ( n , k ) ⊂ ⋯ ⊂ M n .

Thus, { U ( n , k ) : n ∈ N , k ∈ N } is a semi-open cover of G . For each k ∈ N , let V k = U ( 1 , k ) ∪ U ( 2 , k ) ∪ ⋯ ∪ U ( n , k ) . Thus, V k is a semi-open set and V k ⊂ V k + 1 for each k ∈ N .

Let W ( 1 , 1 ) = V 2 ∩ U ( 1 , 2 ) , W ( n , k ) = ( V k + 1 − scl V k − 1 ) ∩ U ( n , k + 1 ) , which holds for 1 ≤ n ≤ k and k > 1 . Thus, W ( n , k ) is semi-open and W ( n , k ) ⊂ U ( n , k + 1 ) ⊂ M n . Suppose that d is a point of G . Let

n 0 = min { n : d ∈ U ( n , k ) , n ∈ N , k ∈ N } , k 0 = min { k : d ∈ U ( n 0 , k ) , k ∈ N } .

Then, d ∈ U ( n 0 , k 0 ) and d is not in U ( n 0 , k 0 − 1 ) . Thus, d is not in scl U ( n 0 , k 0 − 2 ) . Hence, d ∈ U ( n 0 , k 0 ) − scl U ( n 0 , k 0 − 2 ) . Then, d ∈ U ( n 0 , k 0 ) ∩ ( V k 0 + 1 − scl V k 0 − 2 ) and d ∈ W ( n 0 , k 0 − 1 ) . Thus, W = { W ( n , k ) : n ∈ N , k ∈ N } is a semi-open cover of G and we will show that W is star-finite. Suppose W ( n , k ) ∈ W . Then, W ( n , k ) ⊂ U ( n , k + 1 ) ⊂ V k + 1 . Thus, W ( n , k ) ∩ ( V k + 3 − scl V k + 1 ) = ∅ . Hence, W ( n , k ) ∩ ( V k + 3 − scl V k + 1 ) ∩ U ( n , k + 3 ) = ∅ and W ( n , k ) ∩ W ( n , k + 2 ) = ∅ . Then, W ( n , k ) ∩ W ( n , i ) = ∅ and i ≥ k + 2 for each k ∈ N . Thus, W is a star-finite cover.

Since each star-finite semi-open cover has the semi-star property, it follows that there exists a finite set W 0 = { x 1 , x 2 , ⋯ , x t } such that

X = S t ( W 0 , W ) = S t ( x 1 , W ) ∪ ⋯ ∪ S t ( x t , W ) .

Since W is a star-finite cover, it follows that W is point-finite. Thus, S t ( x i , W ) and i = 1 , ⋯ , t is the union of finitely many members of W . Let it be { W i 1 , ⋯ , W i t } . Then, { W 1 1 , ⋯ , W 1 t , W 2 1 , ⋯ , W t t } is a finite subcover of G . Therefore, G is semi-compact.□

The following result is an immediate consequence of Theorem 3.4.

Corollary 3.5

Suppose G is a locally σ semi-compact Ir-topological group and each semi-open set in G is pre-open. If f : G → X is an irresolute surjection, then X is semi-compact.

Corollary 3.6

Suppose G is a locally σ semi-compact Ir-topological group. If each semi-open set in G is pre-open, then each semi-closed set in G is semi-compact.

Corollary 3.7

Suppose G is a locally σ semi-compact Ir-topological group. If each semi-open set in G is pre-open, then G is semi-normal.

Since each T 2 locally compact space is a Tychonoff space [20], we obtain the following theorem directly by Corollary 3.7.

Theorem 3.8

Suppose G is an Ir-topological group, and f : G → Y is a semi-open irresolute bijection. If G is locally semi-compact and each semi-open set in G is pre-open, then Y is a Tychonoff space.

Definition 3.9

[23] A subset A in a topological space X is said to be a regular-open set if and only if A = A ¯ 0 .

It is well known that, a set is regular open if and only if it is semi-closed and pre-open.

Theorem 3.10

Suppose G is an Ir-topological group and each semi-open set in G is pre-open. If A is a locally semi-compact pre-open subgroup, then A is a regular-open Ir-topological subgroup.

Proof

Let us show that A is Ir-topological group. Let f : G × G → G and f 1 : A × A → A be the multiplication mapping. Suppose ( a , b ) in A × A and U is semi-open containing f 1 ( a × b ) = a b in A × A . Since A is pre-open in G , Then, there exists a semi-open set A 0 in G such that U = A ∩ A 0 . Thus, f − 1 ( A 0 ) is semi-open in G × G . Hence, there exist two semi-open sets U a and V b in G such that ( a , b ) ∈ U a × V b ⊂ f − 1 ( A 0 ) ⊂ G × G . Hence, U a ∩ A and V b ∩ A are semi-open in A . Then, f ( ( U a ∩ A ) × ( V b ∩ A ) ) = f 1 ( ( U a ∩ A ) × ( V b ∩ A ) ) = ( U a ∩ A ) ( V b ∩ A ) ⊂ A ∩ A 0 ⊂ U . Thus, f 1 is irresolute.

Let g : G → G and g 1 : A → A be the inverse mapping. Suppose W is semi-open in A . Since A is pre-open in G , it follows that there exists a semi-open set W 0 in G . Then, g − 1 ( W 0 ) is semi-open in G . Hence, g − 1 ( W 0 ) ∩ A = g 1 − 1 ( W ) is semi-open in A and g 1 is irresolute. Therefore, A is an Ir-topological group.

Now, we will show that A is semi-open in G . Suppose l in A . Since A is locally semi-compact, it follows that there exists an open semi-compact neighborhood L in A . According to Corollary 3.7, there exists a semi-open set M in A such that l ∈ M ⊂ scl A M ⊂ L ⊂ A , and scl A M is semi-compact in A , where the set scl A M is defined as the semi-closed set of M in A . Then, scl A M is semi-compact of B . Since G is semi-normal, it follows that scl A M is a semi-closed set in B . Since M ⊂ scl A M , it follows that scl G M ⊂ scl A M , where the set scl G M is defined as the semi-closed set of M in G . Thus, scl G M = scl A M . Since A is pre-open, it follows that there exists a semi-open set N in G such that M = N ∩ A and l in N . Hence, scl G M ⊂ scl G N . Suppose m ∈ scl G N and O is a semi-open neighborhood of m in G . Then, O ∩ N ≠ ∅ . Since each semi-open set in G is pre-open, it follows that O ∩ N is semi-open in G . Then, O ∩ N ∩ A = O ∩ M ≠ ∅ and m in scl G M . Hence, scl G N ⊂ scl G M . Thus,

scl G M = scl G N = scl A M .

Thus, N ⊂ scl G N = scl A M ⊂ A , and A is semi-open in G .

Since the left multiplication l x : G → G is a pre-semi-open mapping, it follows that each left coset l x ( A ) = x A of A is semi-open. Let us show that G − A = ∪ x ∈ G − A x A . Obviously, G − A ⊂ ∪ x ∈ G − A x A . Suppose y in ∪ x ∈ G − A x A . Then, there exists c in A such that y = x c . Thus, c − 1 in A . Assume that y is not in G − A . Then, y c − 1 = x in A , which contradicts the assumption, and y in G − A . Hence, G − A = ∪ x ∈ G − A x A is semi-open and A is semi-closed. Thus, A − 0 ⊂ A . Since A is pre-open, it follows that A ⊂ A − 0 . Therefore, A = A − 0 , and A is regular-open.□

Theorem 3.11

Suppose G is a locally semi-compact Ir-topological group and each semi-open set is pre-open. If Y is an open subgroup of G, then Y is a locally semi-compact Ir-topological group.

Proof

Suppose x is in G . According to Proposition 2.10, Y x is semi-open. Hence, Y 0 = ∪ x ∈ G − Y Y x is semi-open. Then, Y = G − Y 0 is semi-closed.

Suppose a in Y . Then, there exists an open semi-compact neighborhood A containing a . Then, A ∩ Y is open in Y . Let us show that A ∩ Y is semi-closed in Y . Fix b in G − A . Then, there exist two semi-open sets, C a and D a , such that a ∈ C a , b ∈ D a , and C a ∩ D a = ∅ . Then, { C a : a ∈ A } is a semi-open cover of A . Thus, there exists a finite subcover { C a 1 , C a 2 , ⋯ , C a r } of A such that A ⊂ ∪ i = 1 r C a i . Let C = ∪ i = 1 r C a i and D = ∩ i = 1 r D a i . Since each semi-open set in G is pre-open, it follows that D is semi-open and C ∩ D = ∅ . Then, A ∩ D = ∅ . Thus, A is semi-closed in G . Since Y is semi-closed, it follows that A ∩ Y is semi-closed in G . Thus, A ∩ Y is semi-closed in Y .

Now we will show that A ∩ Y is semi-compact in A . Suppose E = { E α : α ∈ I } is a semi-open cover of A ∩ Y in A . Then, there exists an open set F α in A ∩ Y such that F α ⊂ E α ⊂ F α ¯ . Then, there exists an open set G α in A such that F α = G α ∩ A ∩ Y . Hence, E α ¯ ⊂ F α ¯ ⊂ G α ¯ . Let H α = E α ∪ G α . Then, H α ∩ A ∩ Y = E α and H α ¯ = G α ¯ . Hence, H α is semi-open in A . Then, { H α : α ∈ I } ∪ { A − A ∩ Y } is a semi-open cover of A . Thus, there exists a finite semi-open cover

{ H α 1 , H α 2 , ⋯ , H α s } ∪ { A − A ∩ Y } .

Then, there exists a finite semi-open cover { H α 1 , H α 2 , ⋯ , H α s } of A ∩ Y , and A ∩ Y is semi-compact in A .

Suppose { U β : β ∈ j } is a semi-open cover of A ∩ Y in Y . Then, { U β ∩ A : β ∈ j } is a semi-open cover of A ∩ Y in A . Thus, a finite semi-open cover { U β 1 ∩ A , U β 2 ∩ A , ⋯ , U β t ∩ A } of A ∩ Y exists such that A ∩ Y ⊂ ∪ i = 1 t U β i ∩ A ⊂ ∪ i = 1 t U β i . Then, A ∩ Y is semi-compact in Y . Thus, Y is a locally semi-compact space.

Let f 1 : Y × Y → Y and f : G × G → G be the multiplication mapping. Let g 1 : Y → Y and g : G → G be the inverse mapping. Then, f 1 and g 1 are irresolute. The proof is similar to Theorem 3.10 and is omitted. Therefore, Y is a locally semi-compact Ir-topological group.□

We introduce some additional notations for brevity in the following theorems. In [25], the semi-quotient topology was introduced on s-topological groups and irresolute topological groups. This kind of construction will be applied here to topologized groups: Ir-topological groups. Suppose A is an invariant subgroup of an Ir-topological group G and each semi-open is pre-open. Let G ⁄ A = { g A : g ∈ G } and f : G → G ⁄ A , g → g A be the natural projection. Let

B = { B : B ⊂ G ⁄ A , f − 1 ( B ) is semi open in G } .

Let A = { ∅ , G ⁄ A } ∪ B .

Now, we will show that A is a topology. It means that we only need to verify that the intersection of any two sets in B belongs to B . Suppose B 1 and B 2 are subsets in B . Then, f − 1 ( B 1 ∩ B 2 ) = f − 1 ( B 1 ) ∩ f − 1 ( B 2 ) , f − 1 ( B 1 ) and f − 1 ( B 2 ) are semi-open. Thus, there exists an open set C 1 such that C 1 ⊂ f − 1 ( B 1 ) ⊂ C 1 ¯ . Hence, f − 1 ( B 2 ) is pre-open and f − 1 ( B 2 ) ⊂ f − 1 ( B 2 ) ¯ 0 ⊂ f − 1 ( B 2 ) ¯ . Then, f − 1 ( B 1 ) ∩ f − 1 ( B 2 ) ⊂ C ¯ 1 ∩ f − 1 ( B 2 ) ¯ 0 . Take an element x in C ¯ 1 ∩ f − 1 ( B 2 ) ¯ 0 and any neighborhood U x of x . Thus, U x ∩ f − 1 ( B 2 ) ¯ 0 is a neighborhood of x and U x ∩ f − 1 ( B 2 ) ¯ 0 ∩ C 1 ≠ ∅ . Hence, x in C 1 ∩ f − 1 ( B 2 ) ¯ 0 ¯ and C ¯ 1 ∩ f − 1 ( B 2 ) ¯ 0 ⊂ C 1 ∩ f − 1 ( B 2 ) ¯ 0 ¯ . Suppose y ∈ C 1 ∩ f − 1 ( B 2 ) ¯ 0 . Then, y is not in X − f − 1 ( B 2 ) ¯ 0 and there exists an open neighborhood V y of y such that V y ∩ ( X − f − 1 ( B 2 ) ¯ 0 ) = ∅ . Assume that y is not in f − 1 ( B 2 ) . Then, y ∈ f − 1 ( B 2 ) ¯ − f − 1 ( B 2 ) = F r { f − 1 ( B 2 ) } ⊂ F r { f − 1 ( B 2 ) ¯ 0 } . Thus, V y ∩ ( X − f − 1 ( B 2 ) ¯ 0 ) ≠ ∅ . It is contradictory. Then, y ∈ f − 1 ( B 2 ) . Since y ∈ C 1 , it follows that y ∈ f − 1 ( B 1 ) . Thus,

C 1 ∩ f − 1 ( B 2 ) ¯ 0 ⊂ f − 1 ( B 1 ) ∩ f − 1 ( B 2 ) ⊂ f − 1 ( B 2 ) ¯ 0 ∩ C 1 ¯ .

Hence, f − 1 ( B 1 ) ∩ f − 1 ( B 2 ) is semi-open in G ⁄ A . Then, B 1 ∩ B 2 ∈ B and A is a topology on G ⁄ A . We call it the Ir-s-quotient topology, and ( G ⁄ A , A ) is called the Ir-s-quotient topological group.

The following example shows that B is different from the topology C = { C : C ⊂ G ⁄ A , f − 1 ( B ) i s o p e n i n G } .

Example 3.12

Suppose ( X , T ) is a topological space, X = { 1 , 4 , 7 } and

T = { ∅ , X , { 1 } , { 4 } , { 1 , 4 } } .

Then, the semi-open sets of X are T 1 = { ∅ , X , { 1 } , { 4 } , { 1 , 4 } , { 1 , 7 } , { 4 , 7 } } . Let R be a relation from X to X defined by x R y if and only if x ≡ y ( mod 3 ) when x and y are in X . Then, R is an equivalence relation of X . Let R ( 4 ) = { 4 , 7 } . Thus, f − 1 ( R ( 4 ) ) is semi-open and is not open.

It is well known that if f : G → G ⁄ A , g → g A be a natural projection mapping, then f − 1 ( f ( C ) ) = C A for each subset C in G [7], and f is irresolute. The next results are elementary, but important.

Proposition 3.13

Suppose ( G , T ) is an Ir-topological group and each semi-open set is pre-open. If A is an invariant subgroup, then f : ( G , T ) → ( G ⁄ A , A ) is pre-semi-open.

Proposition 3.14

Suppose ( G , T ) is an Ir-topological group and A is a subgroup. If ( G ⁄ A , A ) is semi- T 1 , then A is semi-closed in G .

Theorem 3.15

Suppose G is a locally semi-compact Ir-topological group and each semi-open set is pre-open. If A is an invariant subgroup, then ( G ⁄ A , A ) is a locally semi-compact Ir-topological group.

Proof

Let f : G → G ⁄ A , g → g A and g be the multiplication mapping ( x , y ) → x y from G × G to G , and g 1 be the inverse mapping g → g − 1 from G to G . Then, g and g 1 are irresolute. Let h be the multiplication mapping ( x , y ) → x y from G ⁄ A × G ⁄ A to G ⁄ A , and h 1 be the inverse mapping g → g − 1 from G ⁄ A to G ⁄ A .

Since f ( g ( x , y ) ) = h ( ( f × f ) ( x , y ) ) , it follows that h ( ( f × f ) ( x , y ) ) is irresolute. Assume that a semi-open set D in G ⁄ A such that h − 1 ( D ) is not semi-open. Then, ( h ( f × f ) ) − 1 ( D ) = ( f × f ) − 1 h − 1 ( D ) is semi-open. According to Proposition 3.13, f × f is pre-semi-open. Then,

h − 1 ( D ) = ( f × f ) ( f × f ) − 1 h − 1 ( D )

is semi-open, which contradicts the assumption, and hence h is irresolute. In the same way, h 1 is irresolute. Thus, ( G ⁄ A , A ) is an Ir-topological group.

Suppose a A in G ⁄ A . Then, there exists an open semi-compact neighborhood E of a such that a A in f ( E ) . Thus, f ( E ) is open. Suppose B is a semi-open cover of f ( E ) . Hence, f − 1 ( B ) is a semi-open cover of E . Then, there exists a finite subcover { B 1 , B 2 , ⋯ , B r } , and { f ( B 1 ) , f ( B 2 ) , ⋯ , f ( B r ) } is a semi-open cover of f ( E ) by Proposition 3.13. Therefore, ( G ⁄ A , A ) is a locally semi-compact Ir-topological group.□

4 Conclusion and future direction

In this article, we continued the study of the properties of Ir-topological groups, and some new properties have been obtained. At the same time, we establish the connections between locally semi-compact spaces and Ir-topological groups.

Several directions for future research are discussed below. For example, to obtain different types of topological groups in further research, we suggest adopting an Ir-paratopological group, which has a topology such that multiplication mapping is jointly irresolute instead of an Ir-topological group. The work initiated here is the starting point for continuing work towards that direction and motivating others to do so.

Acknowledgement

Appreciation goes to the referees and the editors for careful reading and suggestions for improvement of this work.

Funding information: This work has been partially supported by the project funded by Beibu Gulf University (Grant nos WDAW201905 and 2023JGA252).
Conflict of interest: The authors have no competing interests to declare that are relevant to the content of this article.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during this study.

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Received: 2023-06-06

Revised: 2023-10-08

Accepted: 2023-10-18

Published Online: 2023-11-20

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

Properties of locally semi-compact Ir-topological groups

Abstract

1 Introduction

2 Preliminaries

Proposition 2.1

Proof

Definition 2.2

Remark 2.3

Definition 2.4

Definition 2.5

Example 2.6

Proposition 2.7

Proof

Definition 2.8

Definition 2.9

Proposition 2.10

Proof

Remark 2.11

Remark 2.12

3 Properties of Ir-topological group

Example 3.1

Proposition 3.2

Example 3.3

Theorem 3.4

Proof

Corollary 3.5

Corollary 3.6

Corollary 3.7

Theorem 3.8

Definition 3.9

Theorem 3.10

Proof

Theorem 3.11

Proof

Example 3.12

Proposition 3.13

Proposition 3.14

Theorem 3.15

Proof

4 Conclusion and future direction

Acknowledgement

References

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