On -strongly -continuity, -openness and (, )-closed graphs in Topological Spaces

Chae et. al. (1995) have studied the concept of α-strongly θ- continuous functions and ( α, θ)-closed graph. The aim of this paper is to investigate several new characterizations and properties of α-strongly θ- continuous functions and (α, θ)-closed graph. Also, we define a new type of functions called αθ-open functions, which is stronger than quasi α-open and hence strongly α-open, and we obtain some characterizations and properties for it. It is shown that the graph of f, G ( f ) is (α, θ)-closed graph if and only if for each filter base Ψ in X θ-converging to some p in X such that f (Ψ) α-converges to some q in Y holds, f ( p) = q.


Introduction
introduced and investigated the concept of -open sets. Chae et.al. (1995) have studied the concept of -strongly -continuous functions. It is shown in Chae et. al. (1995) that the type of -strongly continuous function is stronger than a strongly -continuous function [25] and a strongly -continuous function [12].
The purpose of the present paper is to investigate i) Further characterizations and properties of -strongly -continuous functions [7] and (, )-closed graph [7]. ii) We define a new type of functions called -open functions, which is stronger than quasi -open and hence strongly -open, and we obtain some characterizations and properties for it.

Preliminaries
Throughout the present paper, spaces always mean topological spaces on which no separation axiom is assumed unless explicitly stated. Let E be a subset of a space X. The closure and the interior of E are denoted by Cl(E) and Int(E), respectively. A subset E is said to be regular open (resp. -open [22] and semi-open [16]) if E = Int(Cl(E)) (resp. E  Int(Cl(Int(E))) and E  Cl(Int(E))). A subset E is said to be -open [34] [19] and also the family   of all -open set of X [34], that is,   (called -topology) and   (called an topology) are topologies on X, and it is obvious that        . The family of all -open (resp. -open and feebly-open) set of X is denoted by O(X) (resp. O(X) and FO(X)).
• f : X→Y is called strongly -continuous [27] if for each xX and each open set H of Y containing f (x), there exists an open set G of X containing x such that f (Cl(G))  H.
• f : X→Y is called strongly -continuous [27] if for each open set H of Y, [20] (resp. faintly continuous [18], completely -irresolute [21] and strongly -irresolute  [5] if for each x  X and each • A space X is said to be almost regular[31] if for each regularly closed set R of X and each point x  R, there exist disjoint open sets U and V such that R  U and xV.
• A space X is said to be -Hausdorff [12] if for any x, y  X, x  y, there exist -open sets G and H such that x  G, y  H and G  H = . It is clear that -Hausdorff and Hausdorff are equivalent.
• A space X is said to be -compact [30] (resp. -compact [14]) if and only if every cover of X by -open (resp. -open ) sets has a finite subcover.
• A subset S of a space X is said to be quasi H-closed [28] relative to X if each cover {Ei : iI} of S by open sets of X, there exists a finite subset I of I such that S   {Cl(Ei) : iI}.
• A space X is said to be quasi H-closed [28] if X is quasi H-closed relative to X.

-strongly -continuity
The proof of the following theorem is not hard and therefore, it is omitted.
Proof. This follows from Lemma 3.1 and Theorem 2 of [7].
The proof of the following result directly is true. i) if f is -strongly -continuous and g is -continuous, then g  f is strongly -continuous.

-open
function and we find some characterization and properties for it.     Int ( f (E)), for each subset E of X; c)  Cl( f -1 ( W)), for each subset W of Y.

-open function and E
 X. Since  Int( f (E)).

Functions with (, )-closed graph
In this section we investigate several new properties of (, )-closed graph [7].
. Hence (x, g (y))  (X  Z) \ G ( f ). Since f has (, )-closed graph. Therefore, there exists open set U  X and -open set H  Z containing x and g (y), respectively, such that f (Cl(U))  H = . The -strongly continuity of g implies that there is an open set V of Y such that g (Cl(V))  H. Therefore, we have f (Cl(U))  g (Cl(V)) = . This establishes that (Cl(U)  Cl(V))  E = , which implies that (x, y)  Cl (E). So, E is closed in X  Y. Since H is an -open set containing y, y  Cl( f (E)). Therefore, Cl( f (E)) f (E), which implies that f (E) is -closed.