ALMOST CONTRA REGULAR GENERALIZED b-CONTINUOUS FUNCTIONS

In this paper, the authors introduce a new class of functions called almost contra regular generalized b-continuous function (briefly almost contra rgb-continuous) in topological spaces. Some characterizations and several properties concerning almost contra rgb-continuous functions are obtained. AMS Subject Classification: 54C05, 54C08, 54C10


Introduction
In 2002, Jafari and Noiri introduced and studied a new form of functions called contra-precontinuous functions.The purpose of this paper is to introduce and study almost contra rgb-continuous functions via the concept of rgb-closed sets.Also, properties of almost contra rgb-continuity are discussed.Moreover, we obtain basic properties and preservation theorems of almost contra rgb-continuous functions and relationships between almost contra rgb-continuity and rgb-regular graphs.
Throughout this paper (X, τ ) and (Y, σ) represent the non-empty topological spaces on which no separation axioms are assumed, unless otherwise mentioned.Let A ⊆ X, the closure of A and interior of A will be denoted by cl(A) and int(A) respectively, union of all rgb-open sets X contained in A is called rgb-interior of A and it is denoted by rgb − int(A), the intersection of all rgb-closed sets of X containing A is called rgb-closure of A and it is denoted by rgb − cl(A).

Preliminaries
Definition 1.Let A subset A of a topological space (X, τ ), is called (1) a pre-open set [24] if A ⊆ int(cl(A)).
(4) a generalized bclosed set (briefly gb-closed) [2] if bcl(A) ⊆ U whenever A ⊆ U and U is open in X.
(6) a regular generalized bclosed set (briefly rgb-closed set) [21] if bcl(A) ⊆ U whenever A ⊆ U and U is regular open in X.

Almost Contra Regular Generalized b-ConTinuOus Functions
In this section, we introduce Almost contra regular generalized b-continuous functions and investigate some of their properties.
Then, the following properties hold.
(1) If f is almost contra rgb-continuous and g is an R-map, then g • f is almost contra rgb-continuous.
(2) If f is almost contra rgb-continuous and g is perfectly continuous, then g • f is contra rgb-continuous.
(3) If f is contra rgb-continuous and g is almost continuous, then g • f is almost contra rgb-continuous. Proof.
(1) Let V be any regular open set in Z. Since g is an R-map, ( is rgb-open and rgb-closed set in X. Therefore g • f is continuous and contra rgb-continuous. ( Therefore g is an almost contra rgb-continuous.Proof.Let U be a regular open set in Y .Since f is almost contra rgbcontinuous f −1 (U ) is rgb-closed set in X and X is locally rgb-indiscrete space, which implies f −1 (U ) is an open set in X. Therefore f is almost continuous.Lemma 22.Let A and X 0 be subsets of a space X.
Proof.Let V be any regular open set of Y .By theorem , we have

rgb-Regular Graphs and Strongly
Theorem 27.Let f : (X, τ ) → (Y, σ) be a function and g : (X, τ ) → (X × Y, τ × σ) the graph function defined by g(x) = (x, f (x)) for every x ∈ X.Then f is almost rgb-continuous if and only if g is almost rgb-continuous.
Sufficiency: Let x ∈ X and w be a regular open set of X ×Y containing g(x).There exists U 1 ∈ RO(X, τ ) and

Connectedness
Definition 29.A Space X is called rgb-connected if X cannot be written as a disjoint union of two non-empty rgb-open sets.
Proof.Suppose that Y is not a connected space.Then Y can be written as Theorem 31.The almost contra rgb-continuous image of rgb-connected space is connected.
Proof.Let f : X → Y be an almost contra rgb-continuous function of a rgb-connected space X onto a topological space Y .Suppose that Y is not a connected space.There exist non-empty disjoint open sets V 1 and . This shows that X is not rgb-connected.This is a contradiction and hence Y is connected.Definition 32.A topological space X is said to be rgb-ultra connected if every two non-empty rgb-closed subsets of X intersect.
We recall that a topological space X is said to be hyper connected if every open set is dense.Proof.Suppose Y is weakly Hausdorff.For any distinct points x and y in X, there exist V and W regular closed sets in Y such that f Corollary 36.If f : X → Y is a contra rgb-continuous injection and Y is weakly Hausdorff, then X is rgb − T 1 .
Definition 37. A topological space X is called Ultra Hausdorff space, if for every pair of distinct points x and y in X, there exist disjoint clopen sets U and V in X containing x and y, respectively.Definition 38.A topological space X is said to be rgb − T 2 space if for any pair of distinct points x and y, there exist disjoint rgb-open sets G and H such that x ∈ G and y ∈ H.
Proof.Let x and y be any two distinct points in X.Since f is an injective f (x) = f (y) and Y is Ultra Hausdorff space, there exist disjoint clopen sets U and V of Y containing f (x) and f (y) respectively.Then x ∈ f −1 (U ) and y ∈ f −1 (V ), where f −1 (U ) and f −1 (V ) are disjoint rgb-open sets in X. Therefore X is rgb − T 2 .
Definition 40.A topological space X is called Ultra normal space, if each pair of disjoint closed sets can be separated by disjoint clopen sets.Definition 41.A topological space X is said to be rgb-normal if each pair of disjoint closed sets can be separated by disjoint rgb-open sets.
Theorem 42.If f : X → Y is an almost contra rgb-continuous closed injection and Y is ultra normal, then X is rgb-normal.
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 exists disjoint clopen sets U and V in Y such that f (E) ⊂ U and f (F ) ⊂ V .This implies E ⊂ f −1 (U ) and F ⊂ f −1 (V ).Since f is an almost contra rgb-continuous injection, f −1 (U ) and f −1 (V ) are disjoint rgb-open sets in X.This shows X is rgb-normal.
Theorem 43.If f : X → Y is an almost contra rgb-continuous and Y is semi-regular, then f is rgb-continuous.

Compactness
Definition 44.A space X is said to be: (1) rgb-compact if every rgb-open cover of X has a finite subcover.
(2) rgb-closed compact if every rgb-closed cover of X has a finite subcover.
(3) Nearly compact if every regular open cover of X has a finite subcover.
(4) Countably rgb-compact if every countable cover of X by rgb-open sets has a finite subcover.
(5) Countably rgb-closed compact if every countable cover of X by rgb-closed sets has a finite sub cover.
(6) Nearly countably compact if every countable cover of X by regular open sets has a finite sub cover.
(7) rgb-Lindelof if every rgb-open cover of X has a countable sub cover.
(8) rgb-Lindelof if every rgb-closed cover of X has a countable sub cover.
(9) Nearly Lindelof if every regular open cover of X has a countable sub cover.
(10) S-Lindelof if every cover of X by regular closed sets has a countable sub cover.
(11) Countably S-closed if every countable cover of X by regular closed sets has a finite sub-cover.

Definition 18 .
A function f : X → Y is called weakly rgb-continuous if for each x ∈ X and each open set V of Y containing f (x), there exists U ∈ rgb − O(X; x) such that f (U ) ⊂ cl(V ).Theorem 19.If a function f : X → Y is an almost contra rgb-continuous, then f is weakly rgb-continuous function.Proof.Let x ∈ X and V be an open set in Y containing f (x).Then cl(V ) is regular closed in Y containing f (x).Since f is an almost contra rgb-continuous function by Theorem 3.10 This shows that f is almost weakly rgb-continuous function.Definition 20.A space X is called locally rgb-indiscrete if every rgb-open set is closed in X. Theorem 21.If a function f : X → Y is almost contra rgb-continuous and X is locally rgb-indiscrete space, then f is almost continuous.

Theorem 33 .
If X is rgb-ultra connected and f : X → Y is an almost contra rgb-continuous surjection, then Y is hyper connected.Proof.Suppose that Y is not hyperconnected.Then, there exists an open set V such that V is not dense in Y .So, there exist non-empty regular open subsetsB 1 = int(cl(V )) and B 2 = Y − cl(V ) in Y .Since f is almost contra rgb-continuous, f −1 (B 1) and f −1 (B 2 ) are disjoint rgb-closed.This is contrary to the rgb-ultra-connectedness of X.Therefore, Y is hyperconnected.6.Separation AxiomsDefinition 34.A topological space X is said to be rgb − T 1 space if for any pair of distinct points x and y, there exist a rgb-open sets G and H such that x ∈ G, y / ∈ G and x / ∈ H, y / ∈ H.Theorem 35.If f : X → Y is an almost contra rgb-continuous injection and Y is weakly Hausdorff, then X is rgb − T 1 .

Proof.
Let x ∈ X and V be an open set of Y containing f (x).By definition of semi-regularity of Y , there exists a regular open setG of Y such that f (x) ∈ G ⊂ V .Since f is almost contra rgb-continuous, there exists U ∈ rgb − O(X, x) such that f (U ) ⊂ G. Hence we have f (U ) ⊂ G ⊂ V .This shows that f is rgb-continuous function.