ON gr*-HOMEOMORPHISM IN TOPOLOGICAL SPACES

This paper deals with gr*-closed maps, gr*-open maps, gr*-homeomorphism, gr**-homeomorphism and study their properties. Using these new types of maps, several characterizations and properties have been obtained.


INTRODUCTION
Generalized closed mappings were introduced and studied by Malghan [9].Generalized open maps, rg-closed maps, g*-closed maps, g*-open maps, gpr-closed maps have been introduced and studied by sundaram [15], Arokiarani [1], shiek John [13], and Gnanambal[3] respectively.We give the definitions of some of them which are used our present study.The purpose of this paper is to introduce the concept of new class of maps called gr*-closed maps and gr*-open maps.Further we introduce gr*homeomorphism, gr**-homeomorphism and discuss their properties.

PRELIMINARIES
Definition 2.1 A subset (x,τ) is said to be 1) g-closed [8] set if, cl(A) U whenever A U and U is open in X.
3) gr*-closed [4] if Rcl(A) U whenever A U and U is g-open in X. 4) rg-closed [11] if cl(A) U whenever A U and U is regular open in X.

5) gpr-closed [3] if pcl(A) U whenever A U and U is regular in X.
The complements of the above mentioned closed sets and their respective open sets.Definition 2.2 A map f: X→Y is said to be 3) rg-continous [11] Definition2.3A topological space (X, τ) is said to be i) T 1/2 space if every g closed set is closed.
ii) a T gr* space if every gr*-closed set is closed.[10] if both f and f 1 are g-continuous.
ii) rg-homeomorphism [11] if both f and f 1 are rg-continuous.
(iii) gp*-closed map(briefly gp*-closed) [12], if the image of every closed set in X is gp*-closed in Y.

gr* -CLOSED MAP
(iii) Every gr*-closed map is g-closed map.
(iv) Every gr*-closed map is rg-closed map.
(v) Every gr*-closed map is gpr-closed map.
Proof: Follows form the definition Remark: 3.3 The converse of the above theorem need not be true as seen from the following examples.
Theorem 3.5 A map f: (X,τ)→(Y,σ) is gr*-closed if and only if for each subset S of (Y,σ) and each open set U containing f -1 (s) there is an gr*-open set V of (Y,σ) such that S V and f -1 (V) U.
Proof: Suppose f is gr*-closed set of (X, τ).Let S Y and U be an open set of (X, τ) such that f Conversely, let F be a closed set of (X,τ).Then f -1 (f (F) c )⊂ F c is an open set in (X,τ).By hypothesis, there exists an gr*-open set V in (Y,σ) such that f (F) c V and f -1 (V) F c and so Theorem 3.6 If f: (X,τ)→(Y,σ) is closed map and g: (Y,σ)→(Z,η) is gr*-closed map.Then the composition g∘f: (X, τ)→(Z, η) is gr*-closed map.Proof: Let F be any closed set in(X, τ).Since f is a closed map, f(F) is closed set in (Y,σ).Since g is gr*-closed map, g(f(F)) = g ∘(f(F)) is gr*-closed set in (Z, η).Thus g∘f is gr*-closed map.Remark 3.7 If f: (X, τ)→(Y, σ) is gr*-closed map and g: (Y, σ)→(Z, η) is closed map, then the composition need not be an gr*closed map as seen from the following example.Theorem 3.9 If f: (X, τ)→(Y, σ) and g: (Y, σ)→(Z, η) be two gr*-closed maps where (Y,σ) is T gr* -space.Then the composition g∘f: (X, τ)→(Z, η) is gr*-closed.Proof: Let A be a closed set of (X, τ).Since f is g-closed, f(A) is g-closed in (Y, σ), by hypothesis f(A) is closed.Since g is gr*closed g(f(A)) = g∘(f(A)) is gr*-closed in (Z, η).Thus g∘f is gr*-closed.

Theorem 5.5
Every regular homeomorphism is gr*-homeomorphism, but not conversely.Proof: The proof follows from the theorem 3.2 Example 5.6 Consider X = Y = {a,b,c,d} with topologies τ = {Ф, {d}, {b,c},{b,c,d}, X} and σ = { Ф,{b},{c},{b,c},{b,c,d},Y}.Let f: X→Y be the identity map.Then f is gr*-homeomorphism.But it is not regular homeomorphism.Since the inverse image of the closed set {a} in X is {a} is not closed in Y. Theorem 5.9 Every gr*-homeomorphism is rg-homeomorphism but not conversely.Proof: The proof follows from the definition and fact that every gr*-closed set is rg-closed.
Example 5.10 Consider X = Y = {a,b,c,d} with topology τ = {Ф, {a,c},{b,d},X} and σ = {Ф,{d}, {a,b,c}, Y}.Let f: X→Y be the identity map.Then f is rg-homeomorphism.But it is not gr*-homeomorphism.Since the inverse image of the closed set {a, c} in X is {a, c} is not gr*-closed in Y.
Theorem 5.11 Every gr*-homeomorphism is g-homeomorphism but not conversely.Proof: The proof follows from the definition and fact that every gr*-closed set is g-closed.
Example.5.12 Consider X = Y = {a,b,c,d} with topology τ = {Ф, {a,c},{b,d}, X} and σ = {Ф,{d}, {a,b,c}, Y}.Let f: X→Y be the identity map.Then f is g-homeomorphism.But it is not gr*-homeomorphism.Since the inverse image of the closed set {b, d} in x is {b, d} is not gr*-closed in Y.
Remark 5. 13 The composition of two gr*-homeomorphism need not be a gr*-homeomorphism in general as seen from the following example.