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

S. Malathi , S. Nithyanantha Jothi
Turkish Journal of Analysis and Number Theory. 2018, 6(6), 164-168. DOI: 10.12691/tjant-6-6-4
Received September 22, 2018; Revised October 29, 2018; Accepted December 13, 2018

Abstract

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.

1. Introduction

In 1963, Norman Levine introduced semi-open sets in topological spaces. Also in 1970, he introduced the concept of generalized closed sets. Palaniappan and Rao introduced regular generalized closed (briefly, rg-closed) sets in 1993. In the year 1996, Andrijevic introduced and studied b-open sets. Gnanambal introduced generalized preregular closed (briefly gpr-closed) sets in 1997. N. Levine introduced the concept of semi-continuous function in 1963. In 1980, Jain introduced totally continuous functions. In 1995, T. M. Nour introduced the concept of totally semi-continuous functions as a generalization of totally continuous functions. In 2011, S.S. Benchalli and Umadevi I Neeli introduced the concept of semi-totally continuous functions in topological spaces. In this paper we introduce generalized c*-continuous functions and generalized c*-irresolute functions in topological spaces and study their basic properties.

Section 2 deals with the preliminary concepts. In section 3, generalized c*-continuous functions are introduced and study their basic properties. The generalized c*-irresolute functions in topological spaces are introduced in section 4.

2. 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 regular-closed 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.

Definition: 2.2 A subset A of a topological space X is said to be

i. a clopen set if A is both open and closed in X.

ii. a semi-clopen set if A is both semi-open and semi-closed in X.

Definition: 2.3 6 A subset A of a topological space X is said to be a c*-open set if int(cl(A))⊆A⊆cl(int(A)).

Definition: 2.4 A subset A of a topological space X is called

i. a generalized closed set (briefly, g-closed) 7 if cl(A)⊆H whenever A⊆H and H is open in X.

ii. a regular-generalized closed set (briefly, rg-closed) 8 if cl(A)⊆H whenever A⊆H and H is regular-open in X.

iii. a generalized pre-regular closed set (briefly, gpr-closed) 9 if pcl(A)⊆H whenever A⊆H and H is regular-open in X.

iv. a regular generalized b-closed set (briefly, rgb-closed) 10 if bcl(A)⊆H whenever A⊆H and H is regular-open in X.

v. a regular weakly generalized closed set (briefly, rwg-closed) 11 if cl(int(A))⊆H whenever A⊆H and H is regular-open in X.

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 semi-clopen 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.

Definition: 2.7 21 A function f: X → Y is called an irresolute function if the inverse image of every semi-open subset of Y is semi-open in X.

3. 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 .

Example: 3.2 Let X={a,b,c,d} with topology τ={ϕ, {a}, {b}, {a,b}, {b,c}, {a,b,c}, {a,b,d}, X} and Y={1,2,3} with topology σ={ϕ,{1},Y}. Define f : X → Y by f(a)=f(d)=2, f(b)=3, f(c)=1. Then the inverse image of every closed set in Y is gc*-closed in X. Hence f : X → Y is gc*-continuous.

Proposition: 3.3 Let X,Y be two topological spaces. Then f : X → Y is gc*-continuous if and only if f -1(U) is gc*-open in X for every open set U of Y.

Proof: Suppose f : X → Y is gc*-continuous. Let U be an open set in Y. Then Y∖U is a closed set in Y. This implies, f-1(Y∖U) is a gc*-closed set in X. Since f-1(Y∖U)=X∖f-1(U), we have X∖f-1(U) is a gc*-closed set in X. This implies, f -1(U) is a gc*-open set in X. Conversely, assume that f-1(U) is gc*-open in X for every open set U in Y. Let V be a closed set in Y. Then Y∖V is open in Y. Therefore, f-1(Y∖V) is gc*-open in X. That is, X∖f-1(V) is gc*-open in X. 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 Let X={1,2,3,4} and Y={a,b,c,d,e}. Then, clearly τ={ϕ, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {3,4}, {1,2,3}, {2,3,4}, {1,3,4}, X} is a topology on X and σ={ϕ,{a},{d},{e},{a,d},{a,e}, {d,e},{a,d,e},Y} is a topology on Y. Define f : X → Y by f(1)=b, f(2)=f(3)=d, f(4)=e. Then f is gc*-continuous. Consider the closed set {a,b,c,d} in Y. Then f -1({a,b,c,d})={1,2,3}, 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.

Example: 3.7 Let X={1,2,3,4,5} and Y={a,b,c,d}. Then, clearly τ={ϕ, {1}, {4}, {5}, {1,4}, {1,5}, {4,5}, {1,4,5}, X} is a topology on X and σ={ϕ, {a}, {c}, {d}, {a,c}, {a,d}, {c,d}, {a,c,d}, Y} is a topology on Y. Define f : X → Y by f(1)=f(4)=f(5)=b, f(2)=f(3)=a. Then f is gc*-continuous. Consider the subset {a} in Y. Then f -1({a})={2,3}, which is not a clopen set in X. Therefore, f is not strongly-continuous.

Proposition: 3.8 Let X,Y be two topological spaces. Then every semi-totally continuous function is gc*-continuous.

Proof: Let f: X → Y be a semi-totally continuous function and let V be an open set in Y. Since every open set is semi-open, we have V is semi-open in Y. Therefore, by our assumption, f -1(V) is clopen in X. Then by Proposition 3.7 15, f -1(V) is a gc*-open set in X. Therefore, f : X → Y is gc*-continuous.

The converse of the Proposition 3.8 need not be true as seen from the following example.

Example: 3.9 In Example 3.5, define f : X → Y by f(1)=b, f(2)=f(3)=d, f(4)=e. Then f is gc*-continuous. Consider the semi-open set {d} in Y. Then f -1({d})={2,3}, which is not a clopen set in X. Therefore, f is not semi-totally continuous.

Proposition: 3.10 Let X, Y be two topological spaces. Then every totally-continuous function is gc*-continuous.

Proof: Let f: X → Y be a totally-continuous function and let V be an open set in Y. Then f -1(V) is clopen in X. Therefore, by Proposition 3.7 15, f -1(V) is gc*-open in X. Therefore, f is gc*-continuous.

The converse of the Proposition 3.10 need not be true as seen from the following example.

Example: 3.11 In Example 3.5, define f: X → Y by f(1)=b, f(2)=f(3)=d, f(4)=e. Then f is gc*-continuous. Consider the open set {d} in Y. Then f -1({d})={2,3}, which is not a clopen set in X. Therefore, f is not a totally-continuous function.

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 (gpr-closed) 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.

Example: 3.16 Let X={a, b, c, d, e} and Y={1,2,3,4}. Then, clearly τ={ϕ,{a},{d},{e},{a,d},{a,e}, {d,e}, {a,d,e}, X} is a topology on X and σ={ϕ, {1}, {3}, {4}, {1,3}, {1,4}, {3,4}, {1,3,4}, Y} is a topology on Y. Define f : X → Y by f(a)=1, f(b)=3, f(c)=2, f(d)=f(e)=4. Then f is rg-continuous and gpr-continuous. Consider the closed set {2,4} in Y. Then f -1({2,4})={c,d,e}, which is not a gc*-closed set in X. Therefore f is not gc*-continuous.

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={1,2,3,4} and Y={a,b,c,d,e}. Then, clearly τ={ϕ, {1}, {2}, {1,2}, {1,3}, {1,2,3}, X} is a topology on X and σ={ϕ, {a}, {d}, {e}, {a,d}, {a,e}, {d,e}, {a,d,e}, Y} is a topology on Y. Define f : X → Y by f(1)=a, f(2)=b, f(3)=f(4)=c. Then f is gc*-continuous. Consider the open set {a} in Y. Then f -1({a})={1}, 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.

The strongly semi-continuous functions and gc*-continuous functions are independent. For example, let X={1,2,3,4,5} and Y={a,b,c,d,e}. Then, clearly τ={ϕ, {1}, {4}, {5}, {1,4}, {1,5}, {4,5}, {1,4,5}, X} is a topology on X and σ={ϕ, {a,b}, {c,d}, {a,b,c,d}, Y} is a topology on Y. Define f : X → Y by f(1)=f(2)=a, f(3)=f(5)=c, f(4)=d. Then f is strongly-semi continuous. Consider the closed set {a,b,e} in Y. Then f -1({a,b,e})={1,2}, which is not a gc*-closed set in X. Therefore, f is not gc*-continuous. Now, define g : X → Y by g(1)=g(3)=g(4)=g(5)=e, g(2)=c. Then g is gc*-continuous. Consider the subset {c} in Y. Then g -1({c})={2}, which is not a semi-clopen set in X. Therefore, g is not a strongly-semi continuous function.

The semi-totally semi-continuous functions and gc*-continuous functions are independent. For example, let X={1,2,3,4} and Y={a,b,c,d,e}. Then, clearly τ={ϕ, {1}, {2}, {1,2}, {1,3}, {1,2,3}, X} is a topology on X and σ={ϕ,{a},{d},{e},{a,d},{a,e},{d,e},{a,d,e},Y} is a topology on Y. Define f: X → Y by f(1)=a, f(2)=b, f(3)=f(4)=c. Then f is gc*-continuous. Consider the semi-open set {a,b} in Y. Then f -1({a,b})={1,2}, which is not a semi-clopen set in X. Therefore, f is not a semi-totally semi-continuous function. Now, define g:X → Y by g(1)=g(3)=a, g(2)=g(4)=d. Then g is semi-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.

Proposition: 3.17 Let X, Y be two topological spaces. Then for any bijective function f: X → Y, the following statements are equivalent.

i. f : X → Y is gc*-continuous.

ii. f -1 : Y → X is gc*-open.

Proof: (i) (ii) Assume that f : X → Y is gc*-continuous. Let U be an open set in Y. Since f is gc*-continuous, we have f -1(U) is gc*-open in X. Therefore, f -1 is a gc*-open map. (ii) (i) Assume that f -1 : Y → X is a gc*-open map. Let V be an open subset of Y. By our assumption, f -1(V) is gc*-open in X. Therefore, f is gc*-continuous.

The composition of two gc*-continuous functions need not be gc*-continuous. For example, let X={a,b,c}, Y={1,2,3}, Z={p,q,r}. Then, clearly τ={ϕ, {b}, {c}, {b,c}, X} is a topology on X, σ={ϕ,{1},Y} is a topology on Y and η={ϕ,{p},{p,q},Z} is a topology on Z. Define f: X → Y by f(a)=1, f(b)=3, f(c)=2 and g : Y → Z by g(1)=q, g(2)=p, g(3)=r. Then f and g are gc*-continuous. Consider the closed set {r} in Z. Then (g∘f)-1({r})=f-1(g -1({r}))= f -1({3})= {b}, which is not a gc*-closed set in X. Therefore, g∘f is not a gc*-continuous function.

Proposition: 3.18 Let X,Y and Z be topological spaces. If f : X → Y is gc*-continuous and g: Y → Z is continuous, then g∘f : X → Z is gc*-continuous.

Proof: Let V be a closed set in Z. Since g is continuous, we have g-1(V) is closed in Y. Also, since f is gc*-continuous, we have f -1(g-1(V)) is gc*-closed in X. But f -1(g-1(V))=(g∘f)-1(V). Therefore, (g∘f)-1(V) is gc*-closed in X. Hence g∘f is gc*-continuous.

Proposition: 3.19 Let X,Y and Z be topological spaces. If f : X → Y and g : Y → Z are continuous. Then g∘f : X → Z is gc*-continuous.

Proof: Let V be a closed set in Z. Since g is continuous, we have g-1(V) is closed in Y. Also, since f is continuous, we have f -1(g-1(V)) is closed in Y. That is, (g∘f)-1(V) is closed in Y. By Proposition 4.3 6, (g∘f)-1(V) is gc*-closed in Y. Therefore, g∘f is a gc*-continuous map.

4. Generalized c*-irresolute Functions

In this section, we introduce generalized c*-irresolute functions in topological spaces. Also, we discuss about some of their basic properties.

Definition: 4.1 Let X and Y be two topological spaces. A function f : X → Y is said to be a generalized c*-irresolute (briefly, gc*-irresolute) function if f -1(V) is gc*-closed in X for every gc*-closed set V in Y.

Example: 4.2 Let X={a,b,c,d,e} and Y={1,2,3,4}. Then, clearly τ={ϕ,{a},{d},{e},{a,d},{a,e}, {d,e},{a,d,e},X} is a topology on X and σ={ϕ, {1}, {3}, {4}, {1,3}, {1,4}, {3,4}, {1,3,4}, Y} is a topology on Y. Define f: X → Y by f(a)=f(d)=3, f(b)=f(c)=2, f(e)=4. Then the inverse image of every gc*-closed set in Y is gc*-closed in X. Hence f is gc*-irresolute.

Proposition: 4.3 Let X,Y be two topological spaces. Then f : X → Y is gc*-irresolute if and only if f -1(U) is gc*-open in X for every gc*-open set U of Y.

Proof: Suppose f: X → Y is gc*-irresolute. Let U be an gc*-open set in Y. Then Y∖U is a gc*-closed set in Y. This implies, f -1(Y∖U) is a gc*-closed set in X. Since f -1(Y∖U)=X∖f -1(U), we have X∖f -1(U) is a gc*-closed set in X. This implies, f -1(U) is a gc*-open set in X. Conversely, assume that f -1(U) is gc*-open in X for every gc*-open set U in Y. Let V be a gc*-closed set in Y. Then Y∖V is gc*-open in Y. Therefore, f -1(Y∖V) is gc*-open in X. That is, X∖f -1(V) is gc*-open in X. This implies, f -1(V) is gc*-closed in X. Therefore, f is gc*-continuous.

The irresolute and gc*-irresolute functions are independent. For example, let X={1,2,3, 4,5} and Y={a,b,c,d}. Then, clearly τ={ϕ, {1}, {4}, {5}, {1,4}, {1,5}, {4,5}, {1,4,5}, X} is a topology on X and σ={ϕ, {a}, {c}, {d},{a,c}, {a,d}, {c,d}, {a,c,d}, Y} is a topology on Y. Define f : X → Y by f(1)=f(2)=a, f(3)=f(4)=f(5)=d. Then f is irresolute. Consider the gc*-closed set {a,b} in Y. Then f-1({a,b})={1,2} which is not a gc*-closed set in X. Therefore, f is not gc*-irresolute. Define g : X → Y by g(1)=g(4)=g(5)=b, g(2)=g(3)=a. Then g is gc*-irresolute. Consider the semi-open set {a} in Y. Then g -1({a})={2,3}, which is not a semi-open set in X. Therefore, g is not irresolute.

Proposition: 4.4 Let X,Y be two topological spaces. Then every gc*-irresolute function is rg-continuous.

Proof: Let f : X → Y be a gc*-irresolute function. Let V be a closed set in Y. Then by Proposition 4.3 6, V is gc*-closed set in Y. Since f is gc*-irresolute, f -1(V) is a gc*-closed set in X. Therefore, by Proposition 4.7 6, f -1(V) is a rg-closed set in X. Hence f is rg-continuous.

Proposition: 4.5 Let X,Y be two topological spaces. Then every gc*-irresolute function is gpr-continuous.

Proof: Let f : X → Y be gc*-irresolute and V be a closed set in Y . Then by Proposition 4.3 6, V is gc*-closed in Y. Since f is gc*-irresolute, we have f-1(V) is a gc*-closed set in X. Therefore, by Proposition 4.9 6, f -1(V) is a gpr-closed set in X. Hence f is gpr-continuous.

The converse of Proposition 4.4 and Proposition 4.5 need not be true as seen from the following example.

Example: 4.6 Let X={a,b,c,d,e} with topology τ={ϕ, {a}, {d}, {e}, {a,d}, {a,e}, {d,e}, {a,d,e}, X} and Y={1,2,3,4,5} with topology σ={ϕ,{1,2},{3,4},{1,2,3,4},Y}. Define f: X→ Y by f(a)=2, f(b)=1, f(c)=5, f(d)=3, f(e)=4. Then f is rg-coninuous and gpr-continuous. Consider the gc*-closed set {1,3} in Y. Then f -1({1,3})={b,d}, which is not a gc*-closed set in X. Therefore, f is not gc*-irresolute.

The gc*-irresolute and w-continuous functions are independent. For example,

1. let X={1,2,3,4,5} and Y={a,b,c,d}. Then, clearly τ={ϕ,{1,2},{3,4},{1,2,3,4},X} is a topology on X and σ={ϕ,{a},{c},{d},{a,c},{a,d},{c,d},{a,c,d},Y} is a topology on Y. Define f: X → Y by f(1)=f(4)=a, f(2)=f(3)=b, f(5)=c. Then f is gc*-irresolute. Consider the closed set {a,b,d} in Y. Now, f -1({a,b,d}) ={1,2,3,4}, which is not a w-closed set in X. Therefore, f is not a w-continuous function.

2. let X={1,2,3} and Y={a,b,c}. Then, clearly τ={ϕ, {2}, {3}, {2,3}, X} is a topology on X and σ={ϕ, {a}, {a,b}, {a,c}, Y} is a topology on Y. Define f: X → Y by f(1)=f(3)=b, f(2)=a. Then f is w-continuous. Consider the gc*-closed set {a} in Y. Then f -1({a})={2} is not a gc*-closed set in X. Therefore f is not a gc*-irresolute function.

Proposition: 4.7 Let X,Y be two topological spaces. Then every gc*-irresolute function is gc*-continuous.

Proof: Let f: X → Y be a gc*-irresolute function and V be a closed set in Y. Then by Proposition 4.3 6, V is a gc*-closed set in Y. Since f is gc*-irresolute, we have f -1(V) is a gc*-closed set in X. Therefore, f is gc*-continuous.

The converse of the Proposition 4.7 need not be true as seen from the following example.

Example: 4.8 Let X={1,2,3} and Y={a,b,c}. Then, clearly τ={ϕ, {2}, {3}, {2,3}, 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 gc*-closed set {b} in Y. Then f -1({b})={3}, which is not a gc*-closed set in X. Therefore, f is not a gc*-irresolute function.

Proposition: 4.9 Let X, Y and Z be topological spaces. If f : X → Y and g: Y → Z are gc*-irresolute functions, then g∘f : X → Z is gc*-irresolute.

Proof: Let V be a gc*-closed set in Z . Then g-1(V) is gc*-closed. This implies, f-1(g-1(V)) is gc*-closed. But f-1(g-1(V))= (g∘f)-1(V). Therefore (g∘f)-1(V) is gc*-closed in X. Hence, g∘f is gc*-irresolute.

Proposition: 4.10 Let X,Y and Z be topological spaces. If f: X → Y and g: Y → Z are gc*-irresolute functions, then g∘f: X → Z is gc*-continuous.

Proof: Let f : X → Y and g : Y → Z be two gc*-irresolute functions. Let V be a closed set in Z. By Proposition 4.3 6, V is a gc*-closed set in Z. Then g-1(V) is gc*-closed. This implies, f-1(g-1(V)) is gc*-closed. But f-1(g-1(V))= (g∘f)-1(V). Therefore (g∘f)-1(V) is gc*-closed in X. Hence g∘f : X → Z is gc*-continuous.

Proposition: 4.11 Let X, Y and Z be topological spaces. If f: X → Y is gc*-irresolute and g : Y → Z is continuous, then g∘f: X → Z is gc*-continuous.

Proof: Let V be a closed set in Z . Since g is continuous, we have g-1(V) is closed in Y. By Proposition 4.3 6, g -1(V) is a gc*-closed set in Y. Since f is gc*-irresolute, we have f -1(g-1(V)) is gc*-closed in X. But f-1(g-1(V))=(g∘f)-1(V). Therefore, (g∘f)-1(V) is gc*-closed in X. Hence, g∘f is gc*-continuous.

Proposition: 4.12 Let X,Y and Z be topological spaces. If f: X → Y is gc*-irresolute and g: Y → Z is gc*-continuous, then g∘f : X → Z is gc*-continuous.

Proof: Let V be a closed set in Z. Since g is gc*-continuous, we have g-1(V) is gc*-closed in Y. Since f is gc*-irresolute, we have f-1(g-1(V)) is gc*-closed in X. But fs-1(g-1(V))=(g∘f)-1(V). Therefore, (g∘f)-1(V) is gc*-closed in X. Hence, g∘f is gc*-continuous.

5. 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.

References

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[2]  A.S. Mashhour, M.E. Monsef and S.N. El-Deep, On precontinuous mapping and weak precontinuous mapping, Proc. Math. Phy. Soc. Egypt, 53(1982), 47-53.
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[3]  M. Stone, Application of the theory of Boolean rings to general topology, Trans. Amer. Math. Soc., 41(1937), 374-481.
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[4]  A.I. EL-Maghrabi and A.M. Zahran, Regular generalized-γ-closed sets in topological spaces, Int. Journal of mathematics and computing applications, vol. 3, Nos. 1-2, Jan-Dec 2011, 1-15.
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[5]  D. Andrijevic, On b-open sets, Mat. Vesnik, 48(1996), 59-64.
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[8]  N. Palaniappan, K.C. Rao, Regular generalized closed sets, kyung-pook Math. J., 33(1993), 211-219.
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[11]  A. Vadivel and K.Vairamanickam, rgα-closed sets and rgα-open sets in topological spaces, Int. J. Math. Analysis, 3 (37) (2009), 1803-1819.
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[12]  D. Iyappan and N. Nagaveni, On semi generalized b-closed set, Nat. Sem. on Mat. and comp.sci, Jan(2010), Proc.6.
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[13]  P. Sundaram, M. Sheik John, On w-closed sets in topology, Acta ciencia indica, 4(2000), 389-392.
In article      
 
[14]  M.K.R.S. Veera kumar, On ĝ–closed sets in topological spaces, Bull. Allah. Math. Soc, 18(2003), 99-112.
In article      
 
[15]  S. Malathi and S. Nithyanantha Jothi, On generalized c*-open sets and generalized c*-open maps in topological spaces, Int. J. Mathematics And its Applications, Vol. 5, issue 4-B (2017), 121-127.
In article      
 
[16]  R.C. Jain, The role of regularly open sets in general topological spaces, Ph.D. thesis, Meerut University, Institute of advanced studies, Meerut-India, (1980).
In article      
 
[17]  T. M. Nour, (1995), Totally semi-continuous function, Indian J. Pure Appl. Math., 7, 26, 675-678.
In article      
 
[18]  S.S. Benchalli and U. I Neeli, Semi-totally Continuous function in topological spaces, Inter. Math. Forum, 6 (2011), 10, 479-492.
In article      
 
[19]  Hula M salih, semi-totally semi-continuous functions in topological spaces, AL-Mustansiriya university collage of Educaion, Dept. of Mathematics.
In article      
 
[20]  P. Sundaram and M. Sheik John, Weakly closed sets and weak continuous functions in topological spaces, Proc. 82nd Indian Sci.cong., 49 (1995), 50-58.
In article      
 
[21]  S.G. Crossley and S.K. Hildebrand, Semi-topological properties, Fund. Math., 74 (1972), 233-254.
In article      View Article
 

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S. Malathi, S. Nithyanantha Jothi. On Generalized c*-continuous Functions and Generalized c*-irresolute Functions in Topological Spaces. Turkish Journal of Analysis and Number Theory. Vol. 6, No. 6, 2018, pp 164-168. http://pubs.sciepub.com/tjant/6/6/4
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Malathi, S., and S. Nithyanantha Jothi. "On Generalized c*-continuous Functions and Generalized c*-irresolute Functions in Topological Spaces." Turkish Journal of Analysis and Number Theory 6.6 (2018): 164-168.
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Malathi, S. , & Jothi, S. N. (2018). On Generalized c*-continuous Functions and Generalized c*-irresolute Functions in Topological Spaces. Turkish Journal of Analysis and Number Theory, 6(6), 164-168.
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Malathi, S., and S. Nithyanantha Jothi. "On Generalized c*-continuous Functions and Generalized c*-irresolute Functions in Topological Spaces." Turkish Journal of Analysis and Number Theory 6, no. 6 (2018): 164-168.
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[1]  N.Levine, Semi-open sets and semi-continuity in topological space, Amer. Math. Monthly., 70 (1963), 39-41.
In article      View Article
 
[2]  A.S. Mashhour, M.E. Monsef and S.N. El-Deep, On precontinuous mapping and weak precontinuous mapping, Proc. Math. Phy. Soc. Egypt, 53(1982), 47-53.
In article      
 
[3]  M. Stone, Application of the theory of Boolean rings to general topology, Trans. Amer. Math. Soc., 41(1937), 374-481.
In article      View Article
 
[4]  A.I. EL-Maghrabi and A.M. Zahran, Regular generalized-γ-closed sets in topological spaces, Int. Journal of mathematics and computing applications, vol. 3, Nos. 1-2, Jan-Dec 2011, 1-15.
In article      
 
[5]  D. Andrijevic, On b-open sets, Mat. Vesnik, 48(1996), 59-64.
In article      
 
[6]  S. Malathi and S. Nithyanantha Jothi, On c*-open sets and generalized c*-closed sets in topological spaces, Acta ciencia indica, Vol. XLIIIM, No. 2, 125 (2017), 125-133.
In article      
 
[7]  N. Levine, Generalized closed sets in topology, Rend. Circ. Math. Palermo, 19(2) (1970), 89-96.
In article      View Article
 
[8]  N. Palaniappan, K.C. Rao, Regular generalized closed sets, kyung-pook Math. J., 33(1993), 211-219.
In article      
 
[9]  Y. Gnanambal, On generalized pre regular closed sets in topological spaces, Indian J.Pure Appl. Math., 28(1997), 351-360.
In article      
 
[10]  K. Mariappa, S. Sekar, On regular generalized b-closed set, Int. Journal of Math. Analysis, vol.7, 2013, No.13, 613-624.
In article      
 
[11]  A. Vadivel and K.Vairamanickam, rgα-closed sets and rgα-open sets in topological spaces, Int. J. Math. Analysis, 3 (37) (2009), 1803-1819.
In article      
 
[12]  D. Iyappan and N. Nagaveni, On semi generalized b-closed set, Nat. Sem. on Mat. and comp.sci, Jan(2010), Proc.6.
In article      
 
[13]  P. Sundaram, M. Sheik John, On w-closed sets in topology, Acta ciencia indica, 4(2000), 389-392.
In article      
 
[14]  M.K.R.S. Veera kumar, On ĝ–closed sets in topological spaces, Bull. Allah. Math. Soc, 18(2003), 99-112.
In article      
 
[15]  S. Malathi and S. Nithyanantha Jothi, On generalized c*-open sets and generalized c*-open maps in topological spaces, Int. J. Mathematics And its Applications, Vol. 5, issue 4-B (2017), 121-127.
In article      
 
[16]  R.C. Jain, The role of regularly open sets in general topological spaces, Ph.D. thesis, Meerut University, Institute of advanced studies, Meerut-India, (1980).
In article      
 
[17]  T. M. Nour, (1995), Totally semi-continuous function, Indian J. Pure Appl. Math., 7, 26, 675-678.
In article      
 
[18]  S.S. Benchalli and U. I Neeli, Semi-totally Continuous function in topological spaces, Inter. Math. Forum, 6 (2011), 10, 479-492.
In article      
 
[19]  Hula M salih, semi-totally semi-continuous functions in topological spaces, AL-Mustansiriya university collage of Educaion, Dept. of Mathematics.
In article      
 
[20]  P. Sundaram and M. Sheik John, Weakly closed sets and weak continuous functions in topological spaces, Proc. 82nd Indian Sci.cong., 49 (1995), 50-58.
In article      
 
[21]  S.G. Crossley and S.K. Hildebrand, Semi-topological properties, Fund. Math., 74 (1972), 233-254.
In article      View Article