CONTRA_σgμ - CONTINUOUS FUNCTIONS

In this paper we introduce and discuss some basic properties of contra_σgμ -continuous functions.


INTRODUCTION
The notion αgµ-closed sets in topological spaces was introduced by R. Devi, V. Vijayalakshmi and V. Kokilavani. The concept of Contra continuous mappings was introduced and investigated by J. Dontchev. In this paper we introduce the notion of contra g-continuous functions and discuss their basic properties (a) xker(A) if and only if A  F ≠  for any F  C(X,x).

(b) A  ker(A) and A = ker(A) if A is open in X.
(c) If A  B, then ker(A) ker(B).

Theorem
For a function f : (X, )  (Y, ) the following conditions are equivalent: (1) f is contra g-continuous; (2) for every closed subset F of Y, f −1 (F) gO(X); (3) for each x X and each F  C(Y,f(x)), there exists U gO(X,x) such that f(U)  F; (4) f(gcl(A))  ker(f(A)) for every subset A of X;

Proof.
(1)  (2) Since f is contra g-continuous, inverse image of a closed subset F of Y is gO(X).
(2) (3) It is given F is closed subset of Y and f −1 (F) is gO(X). Hence for xX, there exists U  gO(X) such that f(U)  F.
(3)  (2) Let F be any closed set of Y and x f −1 (F). Then f(x)  F and there exists Ux gO(X,x) such that f(Ux)  F. Therefore, we obtain f −1 (F) = {Ux / xf −1 (F)} and f −1 (F) is g-open.
(2)  (4) Let A be any subset of X. Suppose that y  ker(f (A)). Then by the lemma 3.4, there exists F Therefore, we obtain f(gcl(A))  F =  and y  f(gcl(A)). This implies that f(gcl(A))  ker(f(A).
(4)  (5) Let B be any subset of Y. By lemma 3.4, we have

Theorem
If a function f: (X, )  (Y, ) is contra gcontinuous and Y is regular, then f is gcontinuous.

Corollary
If a function f: (X, )  (Y, ) is contra gcontinuous and Y is regular, and then f is continuous.
We introduce the following definitions

Theorem
If a function f: (X, )  (Y, ) is contra gcontinuous and X is a cTg, then f is g-continuous.

Theorem.
Let X be locally g-indiscrete. If f : (X, )  (Y, ) is contra g-continuous, then f is continuous.  (Njastad, 1965), V be an open set of Y containing f(x) for each x X.

Theorem
Since f is almost g-continuous, there exists U gO(X, x) such that f(U) gint(cl(V)) V. Therefore f(U) ⊆ V.
Conversely, if for each x X and each regular open set V of Y containing f(x), there exists U gO(X, x) such that f(U)  V. This implies V is an open set of Y containing f(x), such that f(U)  V =gint(cl(V)).
Therefore f is almost gcontinuous.

Theorem
If a function f: (X, )  (Y, ) is pre gopen and contra g-continuous, then f is almost g-continuous.

Proof. Let x be any arbitrary point of X and V be an open set containing f(x).Since f is contra g-
. This shows that f is almost g-continuous.

Definition
The graph of a function f: X→Y is said to be contra g-closed if for each (x, y)  (X  Y) -Gr(f), there exists U gO(X, x) and V  C(Y, y) such that (U  V)  Gr (f) = .

Theorem
If f: X →Y is contra g-continuous and Y is Urysohn, then f is Cg-closed in the product space X  Y.

Proof. Let (x, y)  (X  Y) -Gr(f). Then y ≠ f(x) and
there exists open sets A and B such that f(x) A, y  B and cl(A)  cl(B) = . Then there exists V gO(X, x) such that f(V)  cl(A). Therefore, we obtain f(V)  cl(B) = . This shows that f is Cgclosed.

Theorem
If f : X→Y is contra g-continuous with X as locally g-indiscrete then f is continuous.

Proof. Let V be an open set in Y.
Since f is contra g-continuous, f −1 (V) is g-closed set in X. Since X is locally g-indiscrete every g-closed set is open. Hence f −1 (V) is open in X. Therefore f is continuous.

Theorem
If f: X→Y is contra g-continuous and X is cT g-space, then f is contra-continuous.

Proof.
Let V be open set in Y. Since f is contra gcontinuous, f −1 (V) is g-closed in X. Since X is a cT g space, every g-closed set is closed. Hence f −1 (V) closed in X. Therefore f is contra-continuous.

Theorem
If f: X→Y is a surjective pre-closed contra g-continuous with X as cTg space, then Y is locally indiscrete.

Proof. Let V be an open subset in Y. Since f is contra
This means V is closed in X and hence Y is locally indiscrete.

Definition
A space X is said to be g-connected if X cannot be written as a disjoint union of two nonempty g-open sets.

Theorem
A contra g-continuous image of a gconnected space is connected.
Proof. Let f: X→Y be a contra g-continuous map of a g-connected space X on to a topological space Y. If possible, let Y be disconnected. Let A and B form a disconnection of Y. Then A and B are clopen and Y = A  B, where A  B = . Since f is contra gcontinuous map, where f −1 (A) and f −1 B are non-empty gopen sets in X. Also f −1 (A)  f −1 B = . Hence X is not g-connected. This is a contradiction. Therefore Y is connected.

Theorem
If f is contra g-continuous map from a g-connected space X on to any space Y, then Y is not a discrete space.
Proof. Suppose that Y is discrete. Let A be a proper non-empty open and closed subset of Y. Since f is contra g-continuous, f −1 (A) is a proper nonempty g-open and g-closed subset of X, which is a contradiction to the fact that X is g-connected space. Therefore Y is not a discrete space.

Theorem
If f: X→Y is g-irresolute map with Y as locally g-indiscrete space and g : Y→Z is contra