Properties of locally semi-compact Ir-topological groups

: 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 de ﬁ ne a quotient topology on the Ir-topological group and study some topological properties of the space.


Introduction
Recently, topological groups have garnered significant attention from topologists due to their applications in graph algebras and the study of hyperbolic groups [1][2][3][4][5][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][9][10][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][14][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 and the real line is denoted as .The family of all semi-open sets in X is denoted by ( ) X SO .The interior and semi-interior of A in X are denoted as A 0 and ( ) A sop .The closure and semi-closure of A in X are denoted as A and A scl .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].

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

and then there exists a set
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.
[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 x r y a a . Then, = xa ya and = x y.It is contra- dictory, and r a is an injection.For each b in G, there exists a point . Thus, r a is a surjection and r a is bijection.
Define a mapping Therefore, r a is a semi-homeomorphism.□ Properties of locally semi-compact Ir-topological groups  3 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, Ax and xA are semi-open for each x in G.

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.For any is an Ir-topological group which is not a topological group ( f is not continuous, for instance, at ( ) ∈ × G G 3, 3 ).
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.
Obviously, semi-compact space is σ semi-compact, but the converse need not be true.For example, let = X 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 Proof.Since G is a locally σ semi-compact Ir-topological group, it follows that there exists an open neighbor- hood B of the identity e in G such that B is σ semi-compact.Let = ∪ ∈ B B n 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 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 , .Thus, there exist semi-open sets E and F such that e in E, e in F , and and U x0 is a neighborhood of x 0 .Thus, 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 . Thus,  : , . .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 Then, there exists a countable x n n .Arguing by induction, there exists a family , which holds for ≤ ≤ n k , , , , min : , , .
is a semi-open cover of G and we will show that W is star-finite.Suppose Since each star-finite semi-open cover has the semi-star property, it follows that there exists a finite set Since W is a star-finite cover, it follows that W is point-finite.Thus, 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 Since each T 2 locally compact space is a Tychonoff space [20], we obtain the following theorem directly by Corollary 3.7.

Proof. Let us show that
it follows that there exists a semi-open set W 0 in G.Then, ( ) 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 , and M scl A is semi-compact in A, where the set M scl A is defined as the semi-closed set of M in A. Then, Proof.Suppose x is in G.According to Proposition 2.10, of A such that ⊂ ∪ = A C i r a 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.is pre-open and 0 .Take an element x in 0 and any neighborhood U x of x.Thus, ( ) 0 is a neighborhood of x and ( ) Properties of locally semi-compact Ir-topological groups  7 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.

Example 3 . 3 .
Let = X have the standard topology, and

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.

x
is a semi-open cover of O.Then, there exists a semi-open set R y such that ∩ = ∅ R O scl y for each y in P. Let R { } = ∈ R y P : y .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 of O and a countable semi-open cover { } ∈ R n Then, S n and T n are semi-open sets.Hence, ∩ = ∅ S T n m, n in and m in .Let = ∪ ∈ ⊂ P T , and ∩ = ∅ S T

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 semicompact pre-open subgroup, then A is a regular-open Ir-topological subgroup.

GA,..
, where the set M scl G is defined as the semi-closed set of M in G. Thus,= M M scl scl G A .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, ∈ m N scl G 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 M scl G .Hence, and A is semi-open in G. Since the left multiplication → l G G : x is a pre-semi-open mapping, it follows that each left coset ( ) = l A xA x of A is semi-open.Let us show that − = ∪ ∈ − Suppose y in ∪ ∈ − xA x G A .Then, there exists c in A such that = y xc.Thus, − c 1 in A. Assume that y is not in − G A. Then, which contradicts the assumption, and y in − G A. Hence, − = ∪ ∈ − G A xA x G A is semi-open and A is semi-closed.Thus, 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 pre-open.If Y is an open subgroup of G, then Y is a locally semi-compact Ir-topological group.

1 2
Suppose A is an invariant subgroup of an Ir-topological group G and each semi-open is pre-open.Let { 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, are semi-open.Thus, there exists an open set C 1 such that ( ) 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.
Then, there exist two semi-open sets, C a and D a , such that ∈