View of CONCEPTUAL RESEARCH BASED ON HYPER CONNECTED GENERALIZED TOPOLOGY: A REVIEW

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ACCENT JOURNAL OF ECONOMICS ECOLOGY & ENGINEERING

Peer Reviewed and Refereed Journal IMPACT FACTOR: 2.104 (INTERNATIONAL JOURNAL) UGC APPROVED NO. 48767, (ISSN NO. 2456-1037)

Vol. 02, Issue 09,September 2017 Available Online: www.ajeee.co.in/index.php/AJEEE

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CONCEPTUAL RESEARCH BASED ON HYPER CONNECTED GENERALIZED TOPOLOGY:

A REVIEW

Dr. Prakash Deep Agrawal

Department of Mathematics, Govt. Post Graduate College, Kotdwar (Garhwal) Uttarakhand

Abstract- A Csaszar presented and broadly contemplated the thought of summed up open sets. Following Csazar, we present another idea hyper connected. We concentrate some specific properties about associated and hyper connected in summed up topological spaces.

At last, we portray the associated part in summed up topological spaces.

Keywords: Generalized geography, m-structure, powerless design, associated, g-shut, Hyperconnected.

1 INTRODUCTION

The properties of constructions defined by a given set X and a connection, individually relations defined on a class of subsets of X and fulfilling a few conditions are regularly examined. Such constructions are given for instance in [1, 3, 5, 6, 11, 12]. The most popular designs of such kind are topological spaces defined by a conclusion activity.

Summed up topological space is a significant speculation of topological spaces. In the previous decade, Csaszar[4{10] and others have been thinking about summed up topological spaces, and fostering a hypothesis for them. all the more correctly, for the last years, different types of open sets are being contemplated. As of late, a significant commitment to the hypothesis of summed up open sets has been introduced by A.

Csaszar[5{10]. Particularly, the creator defined some fundamental administrators on summed up topological spaces. It is seen that an enormous number of papers are committed to the investigation of summed up open sets like open arrangements of a topological space, containing the class of open sets and having properties pretty much like those of open sets. In the current paper, we present the idea of hyper associated and we concentrated some particular properties about associated and hyper associated in Generalized Topological Spaces. At long last, we describe the associated part in Generalized Topological Spaces.

2. PRELIMINAIRES

Let X be a nonempty set and g be a collection of subsets of X. Then g is called a generalized

topology 2 g. The pair (X, g) is called a

generalized topological space (GTS for short) on X. The elements of g are called g- open sets and their complements are called g- closed sets. We denote the family of all g- closed sets in X by gc(X). The generalized closure of a subset S of X, denoted by cg(S), is the intersection of all generalized closed sets including S. And the generalized interior of S, denoted by ig(S), is the union of generalized open sets contained in S.

Definition 2.1 be a generalized topological space and . Then A is said to be

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Definition 2.2 1Let gX and gY be generalized topologies on X and Y , respectively. Then a

function implies that

Definition 2.3 3Let gX and gY be generalized topologies on X and Y , respectively.

Then a function is said to be

3. ON GENERALIZED CONNECTED SPACES

Definition 3.1 Let (X, gX) be a GTS. X is called: g- connected if there are no nonempty disjoint g-open subsets U; V of X such that _connected if there are no nonempty disjoint open subsets U; V of X such that connected if there are no nonempty disjoint g- semi-open subsets U, V of X such that

connected if there are no nonempty disjoint preopen subsets U; V of X such that U [ V = X g- _-connected if there are no nonempty disjoint -open subsets U; V of X such

that .

Theorem 3.2 Let continuous surjection and let X be connected. Then Y is g- connected.

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Proposition 3.6 Let (X, gX) be a GTS and Then we have the following implcations.

open set And (5) open set

4. ON GENERALIZED HYPERCONNECTED SPACES

Definition 4.1 Let is called g- hyperconnected, if every nonempty g- open subset U of X is g- dense (

g- hyper connected, if every nonempty g - -open subset U of X is gˉˉdense.

hyper connected, if every nonempty semi-open subset U of X is gˉˉdense.

hyper connected, if every nonempty g- preopen subset U of X is gˉˉdense.

hyper connected, if every nonempty open subset U of X is gˉˉense.

Corollary 4.2 Let (X; gX) be a GTS. Then we have the following implcations.

(1) (gˉˉhyperconnected) (4) gˉhyper connected and (5) X is gˉ hyper connected ) (6) gˉˉ- hyper connected ) (7) gˉ hyper connected

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