 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): 20082019
 ISSN (Online) : 22780181
 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 gclosed 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 gclosed(2) (resp. *gclosed(13), g*closed(13), **gclosed(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, gopen, g open, sgopen) set in (X,
Definition 2.02 : A map f : (X, (Y, ) is called gclosed(20) (resp. *gclosed(13), g*closed(24), **gsclosed(8), g closed(12)) map if the image of each closed set in (X, ) is gclosed (resp. *gclosed, g*closed, **gsclosed, 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, *gclosed 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, *gclosed map, g*closed map and g closed map however f is g closed map.
Theorem 3.05: Every g closed map is gclosed map and **gsclosed 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 gclosed map and **gsclosed map however f is g closed map.
Therefore the class of g closed maps properly contains the class of closed maps, the class of *gclosed maps, the class of
g closed maps and the class of g*closed maps and properly contained in class of gclosed maps and the class of **gsclosed 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.
 f1 : (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
 f1 : (Y, (X, ) is g irresolute,
 f is a g *open map and
 f is a g *closed map.
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