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SEMI CONNECTEDNESS IN IRRESOLUTE TOPOLOGICAL GROUPS.

Byline: Rafaqat Noreen, M. Siddique Bosan and Moizud Din Khan

Abstract

In this paper, we will continue the study of irresolute topological groups. We will investigate semi connectedness for irresolute topological groups. It is shown that every semi connected component of identity of an irresolute topological group is equivalent to . Also, a subgroup containing semi open neighbourhood of identity of a semi connected irresolute topological group is equal to.

Keywords: Semi open set, semi closed set, irresolute mapping, semi homeomorphism, irresolute topological group, semi connected space, semi component, semi topological groups with respect to irresoluteness.

1 INTRODUCTION

A topologized group is a triple (Eq.) such that (Eq.) is a group and (Eq.) is a topological space. If both the multiplication (Eq.) and the inversion (Eq.) appings of (Eq.) are continuous, then (Eq.) is called a topological group. If only (Eq.) is continuous, then (Eq.) is called a paratopological group. In case that m is separately continuous, then (Eq.) is called a semi topological group.

If left translations (Eq.) defined by (Eq.) for all (Eq.) are continuous, (Eq.) is called a left semi topological group. Right semi topological groups can be defined similarly. It is known that weaker and less restrictive assumptions can be used to characterize a group topology. In the literature, a lot of research work has been done in this line. In a series of papers, by celebrated mathematicians like A.V. Arhangel'skii, M. Tkachenko, Ljubisa D.R. Kocinac, T.Banakh and Ravsky, on topologized groups emphesisis its importance in the literature. A structure of topologized groups, defined with less restrictive conditions upon continuity is the focus of this paper.

s-topological group [1,2,3] is a generalization of topological groups as well as irresolute topological groups [4].These two topologized groups were obtained by keeping the group and topological structure on a set unaltered, but weakening the continuity conditions in the sense of Levine [5]. In 2014 [2,3] and in 2015 [4], the authors have presented many interesting basic properties of s-topological groups and irresolute topological groups.

The notion of connectedness is a basic, useful and fundamental notion in topological spaces. Das [6] in 1974 defined the weaker form of connectedness and called it semi connected space. Many mathematicians studied semi connected spaces rigorously. In this paper, we investigate some important results related to semi connectedness and semi components in irresolute topological groups.

2 Preliminaries

Throughout this paper, and are always topological spaces on which no separation axioms are assumed. For a subset (Eq.) of a space (Eq.) , the symbols (Eq.) and (Eq.) are used to denote the interior of and the closure of . If (Eq.) is a mapping between topological spaces and and is a subset of (Eq.) , then (Eq.) denotes the pre image of (Eq.) . Our other topological notation and terminology are standard as in (Eq.) [7]. If (Eq.) is a group, then e denotes its identity element, and for a given (Eq.) and (Eq.) denote the left and the right translation by x, respectively. The operation we call the multiplication mapping (Eq.), and the inverse operation is denoted by i.

In 1963, N. Levine [5] defined semi open sets in topological spaces. Since then many mathematicians explored different concepts and generalized them by using semi open sets [8,9, 10, 11]. A subset of a topological space is said to be semi open, if there exists an open set in such that (Eq.) or equivalently, if (Eq.) . denotes the collection of all semi open sets in X, whereas (Eq.) represents the collection of all semi open sets in containing (Eq.).

The complement of a semi open set is said to be semi closed, the semi closure of (Eq.) , denoted by (Eq.) is the intersection of all semi closed subsets of containing [12], [13]. Let us mention that (Eq.) if and only if for any semi open set containing , (Eq.) Clearly, every open (closed) set is semi open (semi closed). It is known that the union of any collection of semi open sets is again a semi open set, while the intersection of two semi open sets need not be semi open. The intersection of an open set and a semi open set is semi open. If (Eq.) and (Eq.) are semi open in spaces (Eq.) and (Eq.) , then (Eq.) is semi open in the product space (Eq.) Basic properties of semi open sets are given in [5], and of semi closed sets and the semi closure in [12,13].

Recall that a set (Eq.) is a semi neighbourhood of a point (Eq.) if there exists (Eq.) such that (Eq.) A set (Eq.) is semi open in (Eq.) if and only if is a semi neighbourhood of each of its points. If a semi neighbourhood of a point is a semi open set, we say that is a semi open neighbourhood of (Eq.).

Definition 1[5] Let (Eq.) and (Eq.) be topological spaces. A mapping (Eq.) is semi continuous, if for each open set in , (Eq.) Clearly, continuity implies semi continuity, (Eq.) the converse need not be true. Notice that a mapping (Eq.) is semi continuous, (Eq.) if and only if for each (Eq.) and each neighbourhood of there is a semi open neighbourhood of x with (Eq.). In [14], Kempisty defined quasi continuous mappings: a (Eq.) mapping (Eq.) is said to be quasi continuous at a point if for each neighbourhood (Eq.) of (Eq.) and each neighbourhood of (Eq.) there is a nonempty open set (Eq.) such that (Eq.) . f is quasi continuous, (Eq.) if it is quasi continuous at each point (see also [15] ). Neubrunnova in [16] proved that semi continuity and quasi continuity coincide.

Definition 2A mapping (Eq.) between topological spaces and is called:

1. semi open [17], if for every open set (Eq.) of , the set (Eq.) is semi open in ;

2. pre semi open [9], if for every semi open set (Eq.) of , the set (Eq.) is semi open in ;

3. irresolute [9], if for every semi open set (Eq.) in , the set (Eq.) is semi open in (Eq.) ;

4. semi homeomorphism [9], if it is bijective, pre semi open and irresolute.

Definition 3A topological space (Eq.) ,is said to be semi compact ([18];[19];[20]); if every semi open cover of has a finite subcover.

Definition 4A subset of a group is symmetric, if (Eq.).

Definition 5 Two non-null subsets of a topological space (Eq.) are said to be semi separated [6], if and only if (Eq.) where denotes the null set.

Definition 6 [6] In topological space (Eq.) , a set which cannot be expressed as the union of two semi separated sets is said to be a semi connected set. The topological space (Eq.) is said to be semi connected, if and only if X is semi connected.

Definition 7 A subspace (Eq.) of a topological space (Eq.) is semi connected [6], if it is semi connected in the subspace topology. i.e if there do not exist disjoint semi open sets (Eq.) and (Eq.) of (Eq.) , such that (Eq.).

Definition 8 Let (Eq.) be a topological space and (Eq.) The semi component [21] of , denoted by (Eq.) is the union of all semi connected subsets of containing Further if (Eq.) and if (Eq.) then the union of all semi connected sets containing and contained in is called the semi component of corresponding to x.

Definition 9 A space is totally semi disconnected, if the singletons are the only semi connected subsets of (Eq.).

Equivalently, a space (Eq.) is totally semi disconnected, if each one-point subset in is its only semi connected component. Of course, every discrete space is totally semi disconnected.

Definition 10 Semi component of an identity element of an irresolute topological group (Eq.) is the largest semi connected subset of that contains the identity element e of the group (Eq.).

Definition 11 [10] Let A be a subset of a space (Eq.) Then a point (Eq.) is said to be a semi isolated point of (Eq.) , if there is a semi open set such that (Eq.).

Definition 12[10] A set is said to be semi discrete, if each point of is semi isolated.

Definition 13 A triple (Eq.) is an irresolute topological group [21] with a group (Eq.) and a topology such that for each (Eq.) and each semi open neighbourhood of (Eq.) there are semi open neighbourhoods of (Eq.) and of such that (Eq.).

Lemma 14 Let (Eq.) be a given mapping. Then is irresolute, if and only if for every (Eq.) and every semi open set (Eq.) containing (Eq.) there exists a semi open set in such that (Eq.) and (Eq.)

Lemma 15 [4] If (Eq.) is an irresolute topological group, then:

1. (Eq.) if and only if (Eq.)

2. If (Eq.) and (Eq.) then (Eq.) and (Eq.) both are semi open in (Eq.).

Lemma 16 [4] For irresolute topological group (Eq.) , every left and right translations are semi homeomorphism.

Lemma 17 [4] Let (Eq.) be any symmetric semi open neighbourhood of (Eq.) in an irresolute topological group (Eq.) . Then the set (Eq.) is a semi open and a semi closed subgroup of (Eq.)

Lemma 18 [4] Let (Eq.) and (Eq.) be subsets of an irresolute-topological group. Then:

(Equations)

Lemma 19 Semi closed subspace of semi compact space is semi compact.

Lemma 20 [21] Let (Eq.) , be open, semi continuous and (Eq.) be open. Then is semi connected, if is semi connected.

Lemma 21[20] If (Eq.) is a semi homeomorphism, then (Eq.) for all

Lemma 22 [20] If (Eq.) is a semi homeomorphism, then (Eq.) for all (Eq.).

Lemma 23 [21] Let be semi connected subspace of (Eq.). Let be subspace of such that (Eq.) . Then is semi connected.

Lemma 24 [21] If the topological space (Eq.) is separated by semi open sets and and if (Eq.) is semi connected subspace of then lies entirely within either (Eq.) or (Eq.).

Lemma 25 semi homeomorphic image of a semi connected space is semi connected.

Lemma 26 [4] Let (Eq.) where is a topological space and is a subspace. Let (Eq.). Then (Eq.)

Lemma 27 [22] A function (Eq.) is pre semi open, if and only if (Eq.) for every subset of (Eq.) .

Lemma 28 [23] Let (Eq.) be a topological space and then (Eq.) .

Theorem 29 If a function

(Equations)

3 Irresolute Topological Groups and Semi connectedness

In this section, we continue the study of irresolute topological groups, then we will present some results on semi connectedness in the presence of irresolute topological groups. The concept of semi connectedness in topological spaces was initially defined and investigated by Das [6] in 1974. With the help of this concept, in this section we will explore the basic properties of irresolute topological groups and see when irresolute topological group terms semi connected irresolute topological group.

We know that a topological group is termed connected, if it satisfies the following equivalent conditions:

1. It is connected as a topological space.

2. and the connected component of the identity element equals the whole group.

For a locally connected topological group, being connected is equivalent to having no proper open subgroup.

Theorem 32 Let (Eq.) be an irresolute topological group, then for each (Eq.) and for every semi open neighbourhood (Eq.) containing (Eq.) there exist semi open neighbouhoods of (Eq.) and (Eq.) of (Eq.) such that

(Equations)

Theorem 33 Let (Eq.) be an irresolute topological group and a subgroup of (Eq.) . If (Eq.) contains a non empty semi open set, then is semi open in (Eq.).

(Equations)

Theorem 34 Let (Eq.) be an irresolute topological group. Then every semi open subgroup of (Eq.) is semi closed in (Eq.).

(Equations)

Theorem 35 A subgroup of an irresolute topological group (Eq.) is semi discrete, if and only if it has a semi isolated point.

(Equations)

Theorem 36 Suppose that (Eq.) is irresolute topological group and (Eq.) be semi compact semi open neighbourhood of (Eq.) . Then there exists a semi open semi compact subgroup of such that (Eq.).

(Equations)

Theorem 37 Let G be an irresolute topological group. Then semi interior of a symmetric subset A of G is again symmetric.

(Equations)

Theorem 38 Let be an irresolute topological group. Then semi closure of a symmetric subset A of G is again symmetric.

(Equations)

Definition 39 A semi topological group with respect to irresoluteness [24] is a group G endowed with a topology such that for each (Eq.) , the translations (Eq.) are irresolute, and such that the inverse mapping (Eq.) is irresolute.

Since every irresolute topological group is semi topological group with respect to irresoluteness, and by (Theorem 4.8, [24] ), we have:

Theorem 40 Let (Eq.) be a semi topological group with respect to irresoluteness and (Eq.) be subsets of G . Then:

1) if is semi compact and B is finite, then (Eq.) and (Eq.) are semi compact.

2) if (Eq.) is semi Lindelof and is countable, then (Eq.) and (Eq.) are semi Lindelof.

Corollary 41 Let (Eq.) be an irresolute topological group and (Eq.) be subsets of (Eq.) . Then:

1) if (Eq.) is semi compact and is finite, then (Eq.) and (Eq.) are semi compact.

2) if (Eq.) is semi Lindelof and is countable, then (Eq.) and (Eq.) are semi Lindelof.

And by (Theorem 4.9,[24] ), we have:

Theorem 42 Let be a semi topological group with respect to irresoluteness, where all the translations are also open. If the semi component (Eq.) of identity is open, then is semi closed normal subgroup.

Corollary 43 Let be an irresolute topological group, where all the translations are also open. If the semi component (Eq.) of identity is open, then is semi closed normal subgroup.

Theorem 44 Let (Eq.) be an irresolute topological group.

Then semi interior of any invariant subgroup of (Eq.) is an irresolute topological invariant subgroup again.

Proof. It follows as the proof of (Theorem 4.6,[24] ).

Theorem 45 Let be an irresolute topological group. Then semi closure of any invariant subgroup of is an irresolute topological invariant subgroup again.

Proof. It follows asthe proof of (Theorem 4.5,[24]).

Theorem 46 Let (Eq.) be an irresolute topological group, (Eq.) the semi component of (Eq.) , and (Eq.) any semi open neighbourhood of (Eq.) Then

(Equations)

Theorem 47 Let (Eq.) be a semi connected irresolute topological group and be a subgroup which contains a semi open neighbourhood of identity, then (Eq.) In particular, a semi open subgroup of a semi connected irresolute topological group G equals G.

(Equations)

Theorem 48 Let (Eq.) be an irresolute topological group and be a semi connected component of (Eq.) , then H is semi closed, and a normal subgroup of (Eq.) .

(Equations)

Theorem 49 Let (Eq.) be a symmetric semi open neighbourhood of the identity element of a semi connected irresolute topological group

(Equations)

Theorem 50 Let (Eq.) be an irresolute topological group and let be a subgroup of (Eq.) . If (Eq.) are semi connected, then (Eq.) is semi connected.

(Equations)

Theorem 51 Let (Eq.) be semi connected semi irressolute topological group and (Eq.) is a totally semi disconnected normal subgroup. Then (Eq.) is contained in the centre of (Eq.).

(Equations)

Theorem 52 Let K be a discrete invariant subgroup of a semi connected irresolute topological group G . Then every element of K commutes with every element of G , that is, K is contained in the center of the group (Eq.).

(Equations)

We have thus proved that the element (Eq.) is in the center of group G. Since x is an arbitrary element of K , we conclude that the center of G contains K.

Theorem 53 Let (Eq.) be an irresolute topological group and let be the semi component of identity in G Then for all (Eq.) is the semi component of a.

(Equations)

Theorem 54 Let G be a semi connected irresolute topological group and e its identity element. If U is any semi open neighborhood of e , then G is generated by U.

(Equations)

Theorem 55 An irresolute topological group G , having a semi open subgroup, is not semi connected.

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