ON g-β-IRRESOLUTE FUNCTIONS ON GENERALIZED TOPOLOGICAL SPACES

In this paper, we introduce and investigate a new kind of function namely g-β-irresolute function along with its two weak and strong forms in generalized topological spaces. Several characterizations and interesting properties of these functions are discussed.


Introduction
Concepts of generalized topological spaces (GTS), generalized open sets and generalized continuity (= (g, g )-continuous functions) were introduced by A. Császár [8,11,14].Since then, several research works devoted to generalize the existing notions of topological spaces to generalized topological spaces have appeared.
On the other hand, Abd El-Monsef et al. [1] introduced the notions of β-open sets and β-continuity in topological spaces early in 1983.Andrijevic [3] introduced the notion of semi-preopen sets which are equivalent to β-open sets.Since then, β-open sets [1] played a significant role in the theory of generalized open sets in topological spaces.In [21], Mahmoud and El-Monsef defined and studied β-irresolute functions.
T. Noiri [26] studied some weak and strong forms of β-irresolute functions in 2003.This work is concerned with the extension various forms of β-irresolute functions to generalized topological spaces.

Preliminaries
A collection g of subsets of X is called a generalized topology (briefly GT) on X [11] if and only if ∅ ∈ g and G i ∈ g for i ∈ I = ∅ implies G = i∈I G i ∈ g.A set X with a GT g on X is called a generalized topological space (GTS) and is denoted by (X, g).By a space X or (X, g), we will always mean a GTS.A GT g on X is called a strong GT [13] if X ∈ g.For a space (X, g), the elements of g are called g-open sets and the complements of g-open sets are called g-closed sets.
For A ⊂ X, the g-closure of A, denoted by cA is the intersection of all g-closed sets containing A and the g-interior of A, denoted by iA is the union of all g-open sets contained in A. It was pointed out in [14] that each of the operations iA and cA are monotonic i.e. if A ⊂ B ⊂ X, then iA ⊂ iB and cA ⊂ cB, idempotent [16], i.e. if A ⊂ X, then i(iA) = iA and c(cA) = cA, iA is restricting [16], i.e. if A ⊂ X, then iA ⊂ A, cA is enlarging [16], i.e., if A ⊂ X, then A ⊂ cA.In a space (X, g), for A ⊂ X, x ∈ iA if and only if there exists an g-open set V containing x such that V ⊂ A and x ∈ cA if and only if In a GTS (X, g), a subset A of X is said to be g-β-open (resp.g-α-open, g-preopen, g-semiopen) [14] if A ⊂ cicA (resp.A ⊂ iciA, A ⊂ icA, A ⊂ ciA).We denote by β(g X ) (resp.α(g X ), π(g X ), σ(g X )) the class of all g-β-open (resp.resp.g-α-open, g-preopen, g-semiopen) sets of (X, g).From [14], it is clear that, forms a GT on X.The complements of a g-β-open sets (resp.g-α-open, g-preopen, g-semiopen) is called g-β-closed (resp.g-α-closed, g-preclosed, g-semiclosed) set.We denote by β(g X , x), the set of all g-β-open sets of (X, g) containing x ∈ X and by βc(g X ) the class of all g-β-closed sets of (X, g).For A ⊂ X, we denote by βcA the intersection of all g-β-closed sets containing A and by βiA the union of all g-β-open sets contained in A.

g-β-regular sets and g-β-θ-open sets
We first state a lemma which will be used in the sequel.Proofs can be checked easily and therefore omitted.
Lemma 3.1.The following hold for a subset A of GTS (X, g): Lemma 3.2.[22] In a GTS (X, g), X is both g-semiopen and g-β-open.Definition 3.1.A subset A of a space X is said to be g-β-regular if it is both g-β-open and g-β-closed.The family of all g-β-regular sets of a space X is denoted by βr(X) and those of containing a point x of X by βr(X, x).Theorem 3.1.For a subset A of a GTS (X, g), (i) A ∈ β(g X ) if and only if βcA ∈ βr(X).
Theorem 3.2.The following are equivalent for a subset A of a GTS (X, g).
We give a proof of the first equality, because that of the other is quite similar.Suppose that, x ∈ βθ-cA.Then there exists, g-β-open set V containing x such that βcV ∩ A = ∅.Therefore by Theorem 3.1, Lemma 3.4.Let A and B be any subset of a GTS (X, g).Then the following properties hold: (viii) A ∈ βr(X) if and only if A is g-β-θ-open and g-β-θ-closed.
Proof: We give only the proofs of (iii) and (iv).Others proofs are obvious.
We now state basic properties of a g-β-irresolute function.Some results of the following Theorem may be analogous to Theorem 3.18 of [24] in terms of other terminologies.
Theorem 4.1.Let f : (X, g X ) → (Y, g Y ) be a function.Then the following are equivalent: (vi) for every x ∈ X and for every g-β-open set V containing f (x), there exists a g-β-open set U of X containing x such that f (U ) ⊂ V ; Proof: (i) ⇒ (ii): Obvious.
Theorem 5.4.For a function f : (X, g X ) → (Y, g Y ), the following are equivalent: (i) f is weakly g-β-irresolute; Proof: The proof is quite similar to the Proof of Theorem 5.1, if we observe that every g-β-closed set is g-β-θ-closed.
Theorem 5.5.For a function f : (X, g X ) → (Y, g Y ), the following are equivalent: (i) f is weakly g-β-irresolute; Proof: The proof is quite similar to Proof of Theorem 5.2 and hence omitted.
Definition 5.1.A GTS (X, g) is said to be g-β-regular if for each F ∈ βc(g X ) and each x / ∈ F , there exist disjoint g-β-open sets U and V such that x ∈ U and F ⊂ V .
Lemma 5.1.The following properties are equivalent in a GTS (X, g): (i) X is g-β-regular; (ii) For each U ∈ β(g X ) and each x ∈ U , there exists V ∈ β(g X ) such that x ∈ V ⊂ βcV ⊂ U ; (iii) For each U ∈ β(g X ) and each x ∈ U , there exists V ∈ βr(X) such that x ∈ V ⊂ U Proof: Follows from Theorem 3.1.
Theorem 5.6.A function f : (X, g X ) → (Y, g Y ) is g-β-irresolute if and only if it is weakly g-β-irresolute and (Y, g Y ) is g-β-regular.
For a subset A of a topological space (X, τ ), the β-closure of A, denoted by βcl(A) is the intersection of all β-open sets containing A and the β-interior of A, denoted by βint(A) is the union of all β-open sets contained in A.

Proposition 5 . 1 .
[28] Let (X, g X ) and (Y, g Y ) be generalized topological spaces and let U = {U × V :U ∈ g X , V ∈ g Y }.Then U generates a generalized topology g X×Y on X × Y , called the generalized product topology on X × Y , i.e. g X×Y = {all possible union of members of U} Proposition 5.2.[28]Let (X, g X ) and (Y, g Y ) be generalized topological spaces, g X×Y be the generalized topology on X × Y , A ⊂ X, B ⊂ Y and K ⊂ X × Y .Then the following hold:(i) K is g X×Y -open if and only if for each (x, y) ∈ K, there exist U x ∈ g X and V y ∈ g Y such that (x, y) ∈ U x × V y ⊂ K.