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Anne Lanham, Lillian Mwanri

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Generalized s-topological Groups

Rehman Jehangir*, Moizud Din Khan*

Department of Mathematics, Preston University Kohat (Islamabad), PakistanDepartment of Mathematics, COMSATS institute of information technology, Chak Shehzad Islamabad, Pakistan

Abstract: In this paper, we explore the notion of generalized semi topological groups. This notion is based upon the two ideas, generalized topological spaces introduced by Csaszar [2,3] and the semi open sets introduced by Levine [7]. We investigate on the notion of generalized topological group introduced by Hussain [4]. We explore the idea of Hussain by considering the generalized semi continuity upon the two maps of binary relation and inverse function.

Keywords: generalized semi open sets, generalized compact sets, generalized continuity, generalized continuity, generalized discrete sets

Citation: Rehman Jehangir, Moizud Din Khan, Generalized s-topological Groups. Turkish Journal of Analysis and Number Theory, 3 (4). 108-110. January 2015. doi:10.12691/tjant-3-4-4

1. Introduction

Let denotes the generalized topological space (, ). In accordance with [3], let be generalized semi open if and only if there exists a generalized open set (-open set) such that , where denotes the generalized closure of the set O in . For more details on generalized topological spaces, we refer to [2,3]. In 2013, Murad et al. [4], defined and studied the concept of generalized topological groups (-topological groups). This study was further extended and published in [5] and [6]. In 2015, C. Selvi and R. Selvi [10] were motivated by -topological groups [4] and S-topological groups [9], and defined on new notion with the name of generalized S-topological groups.

In this paper, we intend to generalized further the notion of -topological groups and -S-topological groups by using -semi continuity. -semi continuity is a generalization of -continuity and it was defined by Á. Császár in [3].

2. Generalized Semi Topological Group

In this section, we will explore the notion of generalized semi topological group. Generalized semi topological groups contains the structure of generalized topology and groups. The whole idea is backed by the generalized semi continuity, as the binary operation and the inverse map undergo the process of generalized semi continuity. We will study the basic definitions and gradual development of the phenomenon.

Lemma 2.1 Let () and () be generalized topological spaces and is generalized semi continuous, then for any subset of , .

Theorem 2.2. Let (, ) and (, ) be generalized topological spaces and let () = be their product generalized topological space if is generalized semi open set in and is generalized semi open in then is generalized semi open in .

Proof. Assume that , where is generalized open in and for . This is nowhere dense set as well. Then,

But is generalized open set in and

Hence,

This proves that is generalized semi open set in .

Theorem 2.3. Let: semi generalized continuous map between two generalized topological spaces. Let be semi generalized compact set relative to then is semi generalized compact in .

Proof. Let be any collection of generalized open set of , such that . Then holds by hypothesis and there exists a finite subset of of such that which shows that is semi generalized compact in .

3. Semi Generalized Topological Group

In this section, we will define semi generalized topological groups (-s-topological groups) and investigate its basic properties.

Definition 3.1. is said to be a - s-topological group if

(1). is generalized topological space;

(2). is a group;

(3). The multiplication map , defined by and the inverse map defined by ; are the generalized semi-continuous. Equivalently, is semi generalized topological group if and for each generalized open set containing , there exist generalized semi open sets containing and containing y, such that, . Since every -continuous is -semi-continuous therefore, every -topological group is -S-topological groups. And -S-topological group may not be a -topological group. Further, we note that, every -topological group is -s-topological group and every -s-topological group is -S-topological group. However, converses may not be true in general. It is evident from the following example:

Example 3.2. Let be the two-element (cyclic) group with the multiplication mapping the usual addition modulo 2. Equip with the Sierpinski topology . Then the collection of all the semi open sets

SO

and that is continuous at , but not continuous at . However, is semi-continuous at . For this, let us take the open set in containing . Then the semi-open set contains . The inverse mapping is continuous and hence semi-continuous. Therefore, is a -s-topological group which is not a topological group. It was noticed in [1] that is not a -S-topological group.

Theorem 3.3. Let be a generalized s-topological group. Let be an inverse mapping defined by ; . Then is generelized semi continuuous mapping.

Proof. Let . Let W be a generalized open set in containing . Then by hypothesis, there exist generalized semi open sets U and V containing e and x, respectively, such that In particular,.

Theorem 3.4. If is semi generalized compact, then is semi generalized compact in a semi generalized topological group .

Proof. Let be a cover of This implies that . This implies that .

Since is semi generalized compact, then there exists a finite set of such that . This implies that That is has a finite subcover of X. Hence is semi generalized compact.

Theorem 3.5. A non empty subgroup of a semi generalized topological group is semi open if and only if its semi interior is non empty.

Proof. Assume that (semi generalized interior). Then by definition there is a semi generalized open set V such that : For every , we have Since V is semi generalized open so is , we conclude that is a semi generalized open set as the union of semi generalized open sets is semi generalized open. Converse of this theorem is quite simple.

Lemma 3.6. Let and be generalized topological spaces and is generalized semi continuous, then for any subset of , .

Further theorem is the extension of the work presented by Bohn Lee [1].

Theorem 3.7. Let be a semi generalized topological group. Then for each generalized open set subset of ; is semi generalized open.

Proof. Let be generalized open in , there exists a generalized open set in , such that,

(By [5])

Because, is semi generalized topological group.

Lemma 3.8. Let be -topological space. If is -semi open and , then is -semi open.

Proof: Let . Since is semi -open therefore there exists a -open set such that .

implies that .

This proves that is semi -open in .

Theorem 3.9. Let be a -s-topological group. Then the multiplication mapping

defined by

is semi -continuous for each

Proof: Let and W be a -open set containing , since X is -semi open sets and containing and , such that

Since is semi -open set containing , therefore, by Theorem-1.6, is -semi open set containing y. Moreover, by Theorem 2, is -semi open set containing . Hence ‘’ is semi -continuous for each .

By Theorems 2.5 and 2.7, it is clear that every -s-topological group is -S-topological group.

Theorem 3.10. Let be a -s-topological group and be any -open set in . Then, for each , both are -semi open in .

Proof: Let . This gives for some .

Since, is -s-topological group, therefore, for -open set A containing , there exist -open set and containing and respectively, such that

Or

This gives This proves that is -semi open set.

Theorem 3.12. Let . be semi -topological group. If is generalized open and , then is generalized semi open in .

Proof. Let and

or, for some

Now, , implies,

Where is generalized open set in , therefore, by the hypothesis, i.e., is semi generalized topological group, there exist generalized semi open set in containing z and containing such that,

or

or

This implies that for each point we can find a generalized semi open set U containing z such that . This means is generalized semi open. Since the union of semi open sets is generalized semi open, therefore,

is generalized semi open.

Lemma 3.13. Let be a semi -topological group and let be the base at identity element e of . Then, for every , there is an element ; so that following holds,

1). .

2). .

3). , for each .

Theorem 3.14. Let be a semi -topological group. Then each left(right) translation is -semi homeomorphic.

Proof. It is obviously bijective map and is semi -continuous containing , there exists -semi open set containing x such that . Again, let be a -open set in , then is semi -open. That is the image of -open set is semi -open. This proves that is -semi homeomorphic.

Note: Let be semi -topological group and . Then for any local base and , then each of the families and is a semi -open neighborhood system of .

References

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[2]   | Á. Császár, Generalized open sets, Acta Math. Hungar., 75 (1997), 65-87.
[3]   | Á. Császár, Generalized topology, generalized continuity, Acta Math.Hungar., 96 (2002), 351-357.
[4]   | M. Hussain, M. Khan and C. Ozel, On generalized topological groups, Filomat, 2013 27(4):567-575.
[5]   | M. Hussain, M. Khan and C. Ozel, More on generalized topological groups, Creative Math. & Inf. 22(2013)(1), 47-51.
[6]   | M. Hussain, M. Khan, A, Z, Özcelik, and C. Ozel, Extension closed properties on generalized topological groups, Arab J Math (2014) 3:341-347.
[7]   | N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly 70 (1963), 36.41.1.
[8]   | Michel Coste, Real Algebraic Sets, March 23, 2005.
[9]   | J. Cao, R. Drozdowski, Z. Piotrowski, Weak continuity properties of topologized groups, Czech. Math. J., 60 (2010), 133-148.
[10]   | C. Silva and R. Silva, On generalized S topological group, International journal of science and research, 6, (2015), 1-4.