**Open Access****Total Downloads**: 16**Authors :**Manoj Garg , Shailendra Singh Rathore**Paper ID :**IJERTV8IS070189**Volume & Issue :**Volume 08, Issue 07 (July 2019)**Published (First Online):**20-08-2019**ISSN (Online) :**2278-0181**Publisher Name :**IJERT**License:**This work is licensed under a Creative Commons Attribution 4.0 International License

#### On ḡ -Closed Mappings in Topological Spaces

Manoj Garg, Shailendra Singh Rathore

- Department and Research Centre of Mathematics, Nehru P. G. College, Chhibramau, Kannauj, U.P., India
Abstract:- In this paper we introduce a new class of closed maps namely g -closed maps which settled in between the class of closed maps and the class of g-closed maps and then we study many basic properties of g -closed maps together with the relationships of some other maps.

2000 Mathematics Subject Classification: 54c10, 54c20.

Key words and phrases: g -closed maps, g *-closed maps

- INTRODUCTION
Malghan(22) and Devi et al(8) introduced the concept of generalized closed maps and semi generalized closed maps respectively in topological spaces. Manoj et al(23) introduced the concept of g -closed sets in topological spaces. In this paper

we introduce a new class of closed maps namely g -closed maps and g *-closed maps.

- PRELIMINARIES
Throughout this paper (X, ), (Y, ) and (Z, ) represent topological spaces on which no separation axioms are assumed unless otherwise mentioned. For a subset A of space (X, the cl(A), int(A) and Ac denote the closure of A, the interior of A and the complement of A in X respectively.

We recall the following definitions:

Definition 2.01: A subset A of a topological space (X, is called g-closed(2) (resp. *g-closed(13), g*-closed(13), **g-closed(17), g -closed(12)) set if cl(A) U (resp. cl(A) U, cl(A) U, scl(A) U, cl(A) U) whenever A U and U is open (resp. g – open, g-open, g -open, sg-open) set in (X,

Definition 2.02 : A map f : (X, (Y, ) is called g-closed(20) (resp. *g-closed(13), g*-closed(24), **gs-closed(8), g -closed(12)) map if the image of each closed set in (X, ) is g-closed (resp. *g-closed, g*-closed, **gs-closed, g -closed) in (Y, ).

- g -CLOSED MAPS In this section we introduce the following definitions.

- INTRODUCTION

- Department and Research Centre of Mathematics, Nehru P. G. College, Chhibramau, Kannauj, U.P., India

Definition 3.01: A map f : (X, (Y, ) is called g -closed (resp. g -open) map if f(A) is g -closed (resp. g -open) set in (Y,

) for every closed (open) set A of (X, .

Definition 3.02: Let (X, ) be a topological space and A X. We define the g -interior of A (briefly g -int(A)) to be the union of all g -open sets contained in A.

Theorem 3.03: Every closed map, *g-closed map, g*-closed map and g -closed map is g -closed map.

Next examples show that the converse of the above theorem is not true in general.

Example 3.04: Let X = Y = {a, b, c}, , {c}, {a, c}, {b, c}, X} and , {a}, {b, c}, Y}. Define f : (X, (Y, ) by f(a) = b, f(b) = a and f(c) = c, then f is not closed map, *g-closed map, g*-closed map and g -closed map however f is g -closed map.

Theorem 3.05: Every g -closed map is g-closed map and **gs-closed map.

Example 3.06: Let X = Y = {a, b, c}, , {a}, {a, b}, {a, c}, X} and , {a}, {a, b}, Y}. Define f : (X, (Y, ) by identity mapping, then f is not g-closed map and **gs-closed map however f is g -closed map.

Therefore the class of g -closed maps properly contains the class of closed maps, the class of *g-closed maps, the class of

g -closed maps and the class of g*-closed maps and properly contained in class of g-closed maps and the class of **gs-closed maps.

Theorem 3.07: If f : (X, (Y, ) be a closed map and g : (Y, (Z, ) be a g -closed map then their composition gof : (X, (Z, ) is g -closed map.

Remark 3.08: The following example shows that the composition of two g -closed maps need not be g -closed map.

Example 3.09: Let X = {a, b, c}, , {a}, {b, c}, X}, , {b}, X} and , {a}, {b}, {a, b}, X}. Define f : (X, (X, ) by f(a) = b, f(b) = c and f(c) = a. Define g : (X, (X, ) by identity mapping then f and g both are g -closed maps but their composition gof : (X, (X, ) is not a g -closed map.

Theorem 3.10: If f : (X, (Y, ) and g : (Y, (Z, ) be two mappings such that their composition gof : (X, (Z, ) be a g -closed map then the following are true

- If f is continuous and surjective, then g is g -closed map.
- If g is g -irresolute and injective, then f is g -closed map.

Theorem 3.11: For any bijective f : (X, (Y, ) the following statements are equivalent.

- f-1 : (Y (X, ) is g -continuous.
- f is g -open map and
- f is g -closed map.

Definition 3.12: A map f : (X, (Y, ) is said to be a g *-closed (resp. g *-open) if the image f(A) is g -closed (resp. g – open) set in (Y, ) for every g -closed (resp. g -open) set A in (X, ).

Theorem 3.13: Every g *-closed map is g -closed map.

The converse is not true in general as it can be seen from the following example.

Example 3.14: Let X = Y = {a, b, c}, , {a}, {b, c}, X} and , {b}, Y}. Define f : (X, (Y, ) by f(a) = b, f(b) = c and f(c) = a then f is g -closed map but not g *-closed map.

Theorem 3.15: For any bijection f : (X (Y, ) the following are equivalent

- f-1 : (Y, (X, ) is g -irresolute,
- f is a g *-open map and
- f is a g *-closed map.

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