On Contra Delta Generalized Pre-Continuous Functions

- In this paper, the notion of contra δgp-continuous functions is introduced by utilizing δgp-closed sets in topological spaces. Some of their fundamental properties are studied the relationships of contra δgp-continuous functions with other related functions are discussed .


I. INTRODUCTION
In 1996,Dontchev [8] initiated the study of contra continuous functions. Subsequently, Jafari and Noiri [15,16] exhibited contra α-continuous and contra pre-continuous functions in topological spaces. In this paper, a new class of generalized contra continuous functions by using δgp-closed sets, called contra δgp-continuous functions is introduced and study some of their basic properties. Relationships between contra δgp-continuous functions and other related functions are investigated.

Definition 2.2
A subset A of a topological space X is called δ-closed [28] if A = cl δ (A) where cl δ (A) = {x ∈ X: int(cl(U))∩A = υ, U ∈ τ and x ∈ U } Definition 2.3 A subset A of a topological space X is called , (i)δgp-closed [5](resp,gpr-closed [13] and gp-closed [17]) if pcl(A) ⊆ U whenever A ⊆ U and U is δ-open (resp, regular open and open) in X.
(ii) gδs-closed [3] if scl(A) ⊆ U whenever A⊆ U and U is δ-open in X.
The complements of the above mentioned closed sets are their respective open sets. Definition 2.4 A function f:X→Y from a topological space X into a topological space Y is called, (i) contra continuous [8] (resp,contra pre-continuous [15], contra-α-continuous [16], contra gp-continuous [7] and contra gpr-continuous) if f −1 (G) is closed (resp, pre-closed, αclosed,gp-closed and gpr-closed) in X for every open set G of Y.
(ii)perfectly-continuous [23] [11] if every open subspace of X is irresolvable. (c)semi-regular [6] if every open set is δ-open in X. (d)Urysohn [29] if for each pair of distinct points x and y of X, there exist open sets U and V containing x and y respectively such that cl(U) ∩ cl(V) = υ. (e)regular [29] if U is open in X and x∈ U, then there is an open set V containing x such that cl(V) ⊆ U.

Definition 2.5
A space X is s a i d t o b e : (i) T δgp -space if every δgp-closed subset of X is closed. (ii) δgpT1/2-space space if every δgp-closed subset of X is pre-closed.

Theorem 3.9
If f:X→Y is a contra δgp-continuous and X is locally δgp-indiscrete space, then f is δgp-continuous. Proof: Let V be a closed set in Y. Since f is contra δgpcontinuous and X is locally δgp-indiscrete space, then f −1 (V) is δgp-closed in X. Hence f is δgp-continuous.

Definition 3.10 [22]
A space X is called locally indiscrete if every open set is closed in X. indiscrete space and f is δgp-continuous, then f −1 (G) is δgp-closed in X. Hence f is contra δgp-continuous.
As a consequence of Theorem 3.12, we have the following Theorem 3.13 and Theorem 3.14.
Theorem 3.13 If f:X→Y is a contra gδs-continuous and X is extremely disconnected space, then f is contra δgpcontinuous.
Theorem 3.14 If f:X→Y is a contra δgp-continuous and X is strongly irresolvable space, then f is contra gδs-continuous.

Theorem 3.15 If f:X→Y is contra δgp-continuous and X is
Tδgp-space,then f is contra continuous. Proof: Suppose X is T δgp -space and f is contra δgpcontinuous. Let G be an open set in Y,by hypothesis f −1 (G) is δgp-closed in X and hence f −1 (G) is closed in X. Therefore f is contra continuous.

Theorem 3.16
If f:X→Y is contra δgp-continuous and X is δgpT 1/2 -space,then f is contra pre-continuous.
Proof: Suppose X is δgpT 1/2 -space and f is contra δgpcontinuous. Let G be an open set in Y,by hypothesis f −1 (G) is δgp-closed in X and hence f −1 (G) is pre-closed in X. Therefore f is contra pre-continuous. Following Theorem is immediate from Lemma 3.18 and Lemma 3.19: The following statements are equivalent for a function f:X→Y: (a)f is perfectly continuous.
(b)f is continuous and contra pre-continuous.
(c)f is continuous and contra gp-continuous.
(d)f is super-continuous and contra δgp-continuous.
(e)f is r-continuous contra gpr-continuous.
(f )f is r-continuous and contra pre-continuous.
(g)f is super-continuous and contra pre-continuous. Recall that for a subset A of a space (X,τ ), the set ∩{U∈τ /A⊆U} is called the kernel of A and is denoted by ker(A).

Lemma 3.23 [14]
The following properties hold for subsets A and B of a space X :

Definition 3.24
A space X is said to be δgpadditive if δGPC(X) is closed under arbitrary intersections.

Theorem 3.25
Let X be δgp-additive, then the following are equivalent for a function f:X→Y.
(ii) For each x ∈ X and each closed set D of Y containing f(x), there exists an δgp-open set C in X containing x such that f(C)⊂D.

Theorem 3.35 [5]
If A⊂X is δgp-closed,then A=δgpcl(A) Remark 3.36 Converse of above theorem is true if X is δgp-additive.

Theorem 3.37
Assume that X is δgp-additive. If f:X→Y and g:X→Y are contra δgp-continuous, X is submaximal and Y is Urysohn. Then F={x∈X:f(x)=g(x)} is δgp-closed in X. Proof: Let x∈X-F, then f(x)=g(x). Therefore,there exist open sets U and V such that f(x)∈U,g(x)∈V and cl(U)∩cl(V)=υ because Y is Urysohn. Since f and g are contra δgp-continuous , f −1 (cl(U)) and g −1 (cl (V))     This is contradiction to the fact that X is δgp-connected. Therefore |K|=1 and hence f is constant.

Definition 3.43
A topological space X is said to be δgp-Hausdorff space if for any pair of distinct points x and y, there exist disjoint δgp-open sets G and H such that x ∈ G and y ∈ H.

Theorem 3.44
If an injective function f:X→Y is contra δgp-continuous and Y is an Urysohn space. Then X is δgp-Hausdorff. Proof: Let x and y be any two distinct points in X and f is injective,then f(x)=f(y). Since Y is an Urysohn space, there exist open sets A and B in Y containing f(x) and f(y) respectively,such that cl(A)∩cl(B)=υ.Then f(x) ∈ cl(A) and f(y) ∈ cl(B). Since f is contra δgpcontinuous, then by Theorem 3.8, there exist δgp-open sets C and D in X containing x and y,respectively,such that f(C) ⊆ cl(A) and f(D) ⊆ cl(B). We have C ∩ D ⊆ f −1 (cl(A)) ∩ f −1 (cl(B)) = f −1 (υ) = υ. Hence X is δgp-Hausdorff.

Definition 3.45 [25]
A space X is called Ultra normal space, if each pair of disjoint closed sets can be separated by disjoint clopen sets.

Definition 3.46
A topological space X is said to be δgpnormal if each pair of disjoint closed sets can be separated by disjoint δgp-open sets.

Theorem 3.47
If f:X→Y be contra δgp-continuous closed injection and Y is ultra normal, then X is δgpnormal. Proof: Let E and F be disjoint closed subsets of X. Since f is closed and injective f(E) and f(F) are disjoint closed sets in Y. Since Y is ultra normal there exist disjoint clopen sets U and V in Y such that f(E)⊂U and f(F)⊂V.
This im-plies E⊂ f −1 (U) and F⊂ f −1 (V). Since f is contra δgp-continuous injection, f −1 (U) and f −1 (V) are disjoint δgp-open sets in X. This shows X is δgp-normal.   Therefore f −1 [g −1 (U)]=(g⋆f ) −1 (U) is δgp-closed in X because f is contra δgp-continuous. Hence g⋆f is contra δgp-continuous. The proofs of (ii) and (iii) are analogous to (i) with the obvious changes. δgp-continuous, f −1 (g −1 (V ))=(g⋆f ) −1 (V ) is δgp-closed set in X. Therefore g⋆f is contra δgp-continuous. Definition 3.52 A function f:X→Y is called pre δgpclosed if the image of every δgp-closed set of X is δgpclosed in Y. Theorem 3.53 Let f:X→Y be pre δgp-closed surject ion and g:Y→Z be a function such that g⋆f:X→Z is contra δgp-continuous,then g is contra δgpcontinuous.