On Slightly ?gb-Continuous Functions

In this paper, we introduce a new class of continuous functions called slightly gb -continuous functions by using gb-closedsets due to Ahmad Al-Obiadi et al.[1]. Ahmad Al-Obiadi et al.[1] introduced the notion of gb-closed sets in a topological space and obtained their various properties. In 2004, Ekici and Caldas[6] introduced the notion of slightly γ-continuity (or slightly b-continuity) which is a weakened form of b-continuity. The relationships of slightly gb-continuity with other weaker forms of continuity viz. weakly gb-continuity, somewhat gb-continuity, almost gb-continuity and faintly gb-continuity have been studied. Throughout the present paper, X and Y are always topological spaces. Mathematics Subject Classification: 2000AMS 54C10

continuity, weakly gb-continuity, faintly gb-continuity, somewhat πgb-continuous. 2. Preliminaries. 2.1.Definition. A subset of a topological space X is said to be 1.regular open [13] if A=int(cl(A)). 2.b-open[3](or γ-open [7]) if A  int(cl(A)) cl(int(A)), 3.gb-closed[1] (resp. g*b-closed[15]) if b-cl(A)  U, whenever A  U and U is open (resp. g-open) in X. 4.g-closed [5] (resp. gb-closed[2], gp-closed[11], gsp-closed[2]) if cl(A)  U (resp. b-cl(A)  U, p-cl(A)  U, sp-cl(A)  U ), whenever A  U and U is -open in X. 5.δ*-open [8] if for each x  A, there exists a clopen subset G of X such that x  G  A. 6.θ-open [14] if for each x  A, there exists an open subset G of X such that x  G  cl (G)  A.

neighbourhood [2] of a point x  X if there exists a gb-open set containing x and is contained in A.

closed, g-closed, gb-closed, gp-closed, gsp-closed, δ*-open ,θ-open) set is called b-open ( resp.

open,gsp-open, δ*-closed ,θ-closed) set. The intersection of all b-closed(resp. δ*-closed) sets of X containing A is called the b- closure ( resp.δ*-closure) of A and denoted by b-cl(A) (resp. δ*-cl(A)). The union of all b-open (resp. δ*- open) subsets of X which are contained in A is called the b-interior (resp. δ*-interior) of A and denoted by b-int(A) (resp. δ*-int(A)).The family of all b-open (resp. b-closed, clopen, b-clopen, δ*-open,δ*-closed, regular open, gb-closed, gb-open) sets in X is denoted by BO(X)(resp.BC(X),CO(X),BCO(X),δ*O(X),δ*C(X),RO(

X),GBC(X),GBO(X)).

2.2.Remark[1]. Every b- closed set is gb-closed. 2.3.Proposition[1]. Every gp-closed set is gb-closed.

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2.5.Remark .We have the following implications for the properties of subsets closed  b-closed g-closed  gp-closed

 

closed  gsp-closed where none of the implications is reversible. 2.6.Example. Let X = { a, b, c } and  = { , X, {a}}. Let A = {c} is a b-closed set but not a closed in X. 2.7.Example. Let X= {a, b, c}, τ = {,{a},X} and A = {a, b}. Then X is the only regular open (-open) set containing A. Hence A is gb-closed, but A is not b-closed, since b-cl (A) = X. 2.8.Example.Let X= {a, b, c}, τ = {  , {a}, {b}, {a, b},X

closed, but A is not gp-closed, since A is regular

3.1.Definition. A function f : X → Y is said to be

πgb-continuous (briefly a.πgb.c.)) if for each x  X and each V  RO(Y) containing f(x), there exists U BO(X)(resp. U  πGBO(X)) containing x such that f(U)  V . 2.weakly b-continuous (briefly w.b.c.) [12] (resp. weakly πgb –continuous (briefly w. πgb.c.)) if for each x  X and each open set VinY containing f(x), there exists U  BO(X) (resp. U  πGBO(X)) containing x such that f(U)  cl(V). 3.somewhat b-continuous (briefly sw.b.c.) [4] (resp. somewhat πgb -continuous (briefly sw. πgb.c.)) if for each open set V in Y and f –1(V )   there exists U  BO(X) (resp. U  πGBO(X)) such U   and U  f –1 (V ). 4. faintly b-continuous (briefly f.b.c.) [10] (resp. faintly πgb -continuous (briefly f.πgb.c.) if for each x  X and each θ-open set V in Y containing f(x), there exists U  BO(X) (resp. U  πGBO(X))) containing x such that f(U)  V . containing f(x), there exists a U  BO(X) (resp. U  πGBO(X)) U  containing x such that f(U)  V . 3.2.Theorem. For a function f : X → Y , the following are equivalent : (a) f is s. gb.c.; (b) f –1(V )  πGBO(X) for every V  CO(X); (c) f –1(V )  πGBC(X) for every V  CO(X); (d) f –1(V )  πGBCO(X) for every V  CO(X). 3.3.Theorem. For a function f : X → Y , the following are equivalent: (a) f is s. gb.c.; (b) f –1(V )  GBO(X) for every δ*-open set V in Y ; (c) f –1(V )  GBC(X) for every δ*-closed set V in Y ; (d) f(b-cl(A))  δ*-cl(f(A)) for every subset A of X; (e) b-cl(f –1 (B))  f –1(δ*-cl(B)) for every subset B of Y . Proof. (a)  (b). Let V be a δ*-open set in Y and let x  f –1(V). Then f(x)  V . The δ*-openness of V gives a U  CO(Y ) such that f(x)  U  V.this implies that x  f –1 (U)  f –1 (V). Since f is s. πgb.c., from Theorem 3.2, we have, f –1(U)  GBO(X). Hence f –1(V) is a gb-neighbourhood of each of its points. Consequently, f –1(V)  GBO(X). (b)  (c). It is obvious from the fact that the complement of a δ*-closed set is δ*-open. (c)  (d). Let A be a subset of X. We have, δ*-cl(f(A)) = ∩{F : f(A)  F, F  δ*C(Y)} is a δ*-closed set in Y. Thus A  f –1(δ*-cl(f(A)) = ∩{ f –1(F) : f(A)  F, F  δ*C(Y)}  GBO(X). Thus, we obtain b-cl(A)  f –1(δ*-cl(f(A)). Hence, f(b-cl(A))  δ*-cl(f(A)). (d)  (e). Let B be a subset of Y . We have f(b-cl(f –1(B))) = δ*-cl(f(f –1(B))) δ*-cl(B) and hence, we obtain, b-cl(f –1(B))  f –1(δ*-cl(B)).

b-cl(f –1(B))  f –1(δ*-cl(B)) = f –1(B). Therefore, f –1 (B) is closed. Hence, by Theorem 3.2, we obtain f is s. πgb.c. 3.4.Theorem. If a function f : X → Y is w. gb.c. then, f is s. gb.c. Proof. Let x  X and let V be a clopen set in Y containing f(x). Therefore,by weakly gb-continuity of f, there exists U  GBO(X) containing x such that f(U)  cl(V ) = V . Since, x  X is arbitrary, hence, f is s. gb.c. 3.5.Remark. The following diagram follows immediately from the definitions in which none of the implications is reversible. f.gb.c  a.gb.c  w.gb.c  s.gb.c 3.6.Example. Let X = Y = {a, b, c}, τ = {, {a}, X }, σ = {, {a}, {c}, {a, c}, Y}. Then the identity function i : (X, τ) → (Y, σ) is s.πgb.c. but not w. gb.c. at b  X. 3.7.Remark. From definition, it is clear that every a.

3.8.Definition. A toipological space X is said to be extremally disconnected [16] if closure of every open set is open in X. 3.9.Theorem If a function f : X → Y is f. gb.c. then, f is s. gb.c. Proof. The result is obvious from the fact that every clopen set is θ-open. 3.10.Remark. The converse of the above result is however, in general, not true as shown by the following example. 3.11.Example. Let τ = {G  R : 0  G}  {} and let σ be the usual topology on R. Then the identity function i : (R, τ) → (R, σ) is s. gb.c. but not f.πgb.c. at all points of R except 0. Proof. Let x  X and let V be a θ-open set in Y containing f(x). Thus there exists an open set W such that f(x)  cl(W)  V . By extremally disconnectedness of Y, cl(W) is open. Thus, cl(W)  CO(Y ). Since, f is s. gb.c., therefore, there exists a b-open set U containing x such that f(U)  cl(W)  V . Since, x  X is arbitrary, therefore, f is f. gb.c. 3.13.Theorem. Let f : X → Y be a function, where, Y is extremally disconnected. Then f is f. gb.c. if and only if f is s. gb.c. Proof. It can be directly obtained by using Theorem 3.9 and Theorem 3.12. 3.14.Remark. Somewhat b-continuity and slightly gb-continuity are independent of each other as shown by 3.15.Example. The function defined in Example 3.11 is s. gb.c. but not sw. gb.c. Again let X = Y = {a, b, c}, τ = {, {a}, {b}, {a, b}, {a, c},X}, σ = {, {a},{b, c}, Y}. Then the identity function i : (X, τ) → (Y, σ) is sw. gb.c. but not s. gb.c.

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