On Generalized c*-continuous Functions and Generalized c*-irresolute Functions in Topological Spaces

The aim of this paper is to introduce the notion of generalized c*-continuous functions and generalized c*-irresolute functions in topological spaces and study their basic properties.


Preliminaries
Throughout this paper X denotes a topological space on which no separation axiom is assumed. For any subset A of X, cl(A) denotes the closure of A, int(A) denotes the interior of A, pcl(A) denotes the pre-closure of A and bcl(A) denotes the b-closure of A. Further X∖A denotes the complement of A in X. The following definitions are very useful in the subsequent sections. Definition: 2.1 A subset A of a topological space X is called i. a semi-open set [1] if A⊆cl(int(A)) and a semi-closed set if int(cl(A))⊆A. ii. a pre-open set [2] if A⊆int(cl(A)) and a pre-closed set if cl(int(A))⊆A. iii. a regular-open set [3] if A=int(cl(A)) and a regularclosed set if A=cl(int(A)). iv. a γ-open set [4] (b-open set [5]) if A⊆cl(int(A)) ∪int(cl(A)) and a γ-closed set (b-closed set) if int(cl(A))∩cl(int(A))⊆A.

Turkish Journal of Analysis and Number Theory
vi. a semi-generalized b-closed set (briefly, sgb-closed) [12] if bcl(A)⊆H whenever A⊆H and H is semi-open in X. vii. a weakly closed set (briefly, w-closed) [13] (equivalently, ĝ-closed [14]) if cl(A)⊆H whenever A⊆H and H is semi-open in X. The complements of the above mentioned closed sets are their respectively open sets. Definition: 2.5 [6] A subset A of a topological space X is said to be a generalized c*-closed set (briefly, gc*-closed set) if cl(A)⊆H whenever A⊆H and H is c*-open. The complement of the gc*-closed set is gc*-open [15]. Definition: 2.6 A function f : X → Y is called i. semi-continuous [1] if the inverse image of each open subset of Y is semi-open in X. ii. totally-continuous [16] if the inverse image of every open subset of Y is clopen in X. iii. strongly-continuous [3] if the inverse image of every subset of Y is clopen subset of X. iv. totally semi-continuous [17] if the inverse image of every open subset of Y is semi-clopen in X . v. strongly semi-continuous [17] if the inverse image of every subset of Y is semi-clopen in X . vi. semi-totally continuous [18] if the inverse image of every semi-open subset of Y is clopen in X. vii. semi-totally semi-continuous [19] if the inverse image of every semi-open subset of Y is semiclopen in X. viii. rg-continuous [9] if inverse image of every closed subset of Y is rg-closed in X. viii. gpr-continuous [9] if inverse image of every closed subset of Y is gpr-closed in X. ix. w-continuous [20] (ĝ-continuous [14]) if inverse image of every closed subset of Y is w-closed in X.

Generalized c*-continuous Functions
In this section, we introduce generalized c*-continuous functions and study its basic properties. Now, we begin with the definition of generalized c*-continuous function. Definition: 3.1 Let X and Y be two topological spaces. A function f: X → Y is called a generalized c*-continuous (briefly, gc*-continuous) function if f -1 (V) is gc*-closed in X for every closed set V of Y .
This implies, f -1 (V) is gc*-closed in X. Therefore, f is gc*-continuous. Proposition: 3.4 Let X, Y be two topological spaces. Then every continuous function is gc*-continuous. Proof: Let f: X → Y be a continuous function. Let V be a closed set in Y. Then f -1 (V) is a closed set in X. By Proposition 4.3 [6], f -1 (V) is gc*-closed in X. Therefore, f is gc*-continuous.
The converse of the Proposition 3.4 need not be true as seen from the following example. Example: 3. 5 , which is not a closed set in X. Therefore, f is not continuous. Proposition: 3.6 Let X, Y be two topological spaces. Then every strongly-continuous function is gc*-continuous.
Proof: Let f: X → Y be a strongly-continuous function and let V be a closed set in Y. Then f -1 (V) is a clopen set in X. This implies, f -1 (V) is closed in X. By Proposition 4.3 [6], f -1 (V) is gc*-closed. Therefore, f : X → Y is gc*-continuous.
The converse of the Proposition 3.6 need not be true as seen from the following example. Proposition: 3.12 Let X,Y be two topological space. Then every w-continuous (ĝ-continuous) function is gc*-continuous. Proof: Let f: X → Y be a w-continuous function. Let V be a closed set in Y. Then f -1 (V) is w-closed in X. By Proposition 4.5 [6], we have f -1 (V) is gc*-closed in X. Therefore, f is gc*-cotinuous.
The converse of the Proposition 3.12 need not be true as seen from the following example. Example: 3.13 Let X={1,2,3} and Y={a,b,c}. Then, clearly τ={ϕ ,{2},{1,2},X} is a topology on X and σ={ϕ ,{a},Y} is a topology on Y. Define f :X → Y by f(1)=a, f(2)=c, f (3)=b. Then f is gc*-continuous. Consider the closed set {b,c} in Y. Then f -1 ({b,c})={2,3} which is not a w-closed set in X. Therefore, f is not a w-continuous function. Proposition: 3.14 Let X, Y be two topological spaces. Then every gc*-continuous function is rg-continuous. Proof: Letf: X → Y be a gc*-continuous function. Let V be a closed set in Y. Then f -1 (V) is a gc*-closed set in X. By Proposition 4.7 [6], we have f -1 (V) is rg-closed (gprclosed) in X. Therefore, f is rg-continuous. Proposition: 3.15 Let X,Y be two topological spaces. Then every gc*-continuous function is gpr-continuous.
Proof: Let f: X → Y be a gc*-continuous function. Let V be a closed set in Y. Then f -1 (V) is a gc*-closed set in X. By Proposition 4.9 [6], we have f -1 (V) is gpr-closed in X. Therefore, f is gpr-continuous.
The converse of Proposition 3.14 and Proposition 3.15 need not be true as seen from the following example. The semi-continuous functions and gc*-continuous functions are independent. For example, In Example 3.7, define f : X → Y by f(1)=f(2)=a, f(3)=b, f(4)=f (5)=d. Then f is semi-continuous. Consider the closed set {b,c,d} in Y. Then f -1 ({b,c,d})={3,4,5}, which is not a gc*-closed set in X. Therefore, f is not a gc*-continuous. Now, define g: X → Y by g(1)=g(4)= g(5)=b, g(2)=g(3)=a. Then g is gc*-continuous. Consider the closed set {b,d} in Y. Then g -1 ({b,d})={1,4,5}, which is not a semi-closed set in X. Therefore, g is not a semi-continuous function.
The totally-semi continuous functions and gc*-continuous functions are independent. For example, let X={1,2,3, , which is not a semi-clopen set in X. Therefore, f is not a totally-semi continuous function. Now, define g: X → Y by g(1)=g(3)=a, g(2)=g(4)=d. Then g is totally-semi continuous. Consider the closed set {a,b,c} in Y. Then g -1 ({a,b,c})={1,3}, which is not a gc*-closed set in X. Therefore, g is not a gc*-continuous function.

Conclusion
In this paper we have introduced gc*-continuous and gc*-irresolute functions in topological spaces and studied some of their basic properties. Also we have studied the relationship between gc*-continuous functions and some of the functions already exist.