On ḡ -Closed Mappings in Topological Spaces

On ḡ -Closed Mappings in Topological Spaces

Manoj Garg, Shailendra Singh Rathore

1. 1. 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

      1. 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.

      2. 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, ).

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

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

1. If f is continuous and surjective, then g is g -closed map.
2. 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.

1. f-1 : (Y (X, ) is g -continuous.
2. f is g -open map and
3. 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

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

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