ON WG-CONTINUOUS FUNCTIONS IN ASSOCIATED WEAK SPACES

The purpose of this paper is to introduce the notions of wg-continuity and wg∗continuity by defined wgτ -open sets in associated w-spaces, and to study some properties and the relationships among such notions and the other continuity. AMS Subject Classification: 54A05, 54B10, 54C10, 54D30


Introduction
Siwiec [18] introduced the notions of weak neighborhoods and weak base in a topological space.We introduced the weak neighborhood systems defined by using the notion of weak neighborhoods in [8].The weak neighborhood system induces a weak neighborhood space which is independent of neighborhood spaces [2] and general topological spaces [1].The notions of weak structure, w-space, W -continuity and W * -continuity were investigated in [9].In fact, the set of all g-closed subsets [3] in a topological space is a kind of weak structure.
In [11], we studied the notion of generalized w-closed sets (simply, gw-closed sets) in a w-space.
In [10], we introduced the notion of an associated weak space (simply, associated w τ -space) containing a given topology τ , and moreover, we studied the notion of generalized w τ -closed sets (simply, gw τ -closed sets) [14] in an associated w τ -space with a topology τ .
In [12], we introduced the notions of gw τ -continuous and gw * τ -continuous functions in associated w-spaces, and studied their characterizations and the relationships among them.
In [15], we studied w τ -generalize closed sets (simply, wg-closed) in an associated weak space w τ in the similar way introduced by Levine [3] in topological spaces.
In this paper, we are going to introduce the notions of wg-continuity and wg * -continuity in associated w-spaces by defined wg-open sets, and to study some properties and the relationships among such notions and the other continuity in associated w-spaces.

Preliminaries
Definition 2.1 ( [9]).Let X be a nonempty set.A subfamily w X of the power set P (X) is called a weak structure on X if it satisfies the following: (1) ∅ ∈ w X and X ∈ w X . ( Then the pair (X, w X ) is called a w-space on X.Then V ∈ w X is called a w-open set and the complement of a w-open set is a w-closed set.
The collection of all w-open sets (resp., w-closed sets) in a w-space X will be denoted by W (X) (resp., W C(X)).We set Let S be a subset of a topological space X.The closure (resp., interior) of S will be denoted by clS (resp., intS).A subset S of X is called a preopen set [6] (resp., α-open set [17], semi-open [4]) if S ⊂ int(cl(S)) (resp., S ⊂ int(cl(int(S))), S ⊂ cl(int(S))).The complement of a preopen set (resp., αopen set, semi-open) is called a preclosed set (resp., α-closed set, semi-closed).The family of all preopen sets (resp., α-open sets, semi-open sets) in X will be denoted by P O(X) (resp., α(X), SO(X)).We know the family α(X) is a topology finer than the given topology on X.
Moreover, a subset S of X is said to be g-closed [3] Then the family GO(X) weak structures on X.But P O(X), GP O(X) and SO(X) are not weak structures on X.A subfamily m X of the power set P (X) of a nonempty set X is called a minimal structure on X [5] if ∅ ∈ w X and X ∈ w X .Thus clearly every weak structure is a minimal structure.
Let (X, w X ) be a w-space.For a subset A of X, the w-closure of A and the w-interior of A are defined as follows: (1) ).Let (X, w X ) be a w-space and A ⊆ X.Then the following things hold: (1) If A ⊆ B, then wI(A) ⊆ wI(B); wC(A) ⊆ wC(B).
(2) wI(wI(A)) = wI(A); wC(wC(A)) = wC(A). ( Let X be a nonempty set and let (X, τ ) be a topological space.A subfamily w of the power set P (X) is called an associated weak structure (simply, w τ ) [10] on X if τ ⊆ w and w is a weak structure.Then the pair (X, w τ ) is called an associated w-space with τ .
Let (X, w τ ) be an associated w-space with a topology τ and A ⊆ X.Then A is called a w τ -generalized closed set (simply, wg-closed set) [15] We recall that: A is called a generalized closed set (simply, g-closed set) [3] if cl(A) ⊆ U , whenever A ⊆ U and U is open.Then if w τ = τ , then a w τ -generalized closed set is exactly a g-closed set.In general, the intersection of two wg-closed sets is not wg-closed and the union of two wg-open sets is not wg-open.
Let (X, w τ ) be an associated w-space with a topology τ .For a subset A of X, w τ g-closure of A and w τ g-interior of A are defined as the following: (1) w τ gC(A) = ∩{F : A ⊆ F, F is w τ g-closed}.
Obviously we obtain the following theorem: Let (X, w τ ) be an associated w-space with a topology τ and A ⊆ X.Then A is called a generalized w τ -closed set (resp., a generalized w-closed set ) (simply, gw τ -closed set [14]) (resp., gw-closed set) if cl(A) ⊆ U (resp., wC(A) ⊆ U ), whenever A ⊆ U and U is w-open.Then since τ ⊆ w, every gw τ -closed set is gw-closed, and every gw-closed set is w τ g-closed.But, the converses are not be true in general.
We recall that: Let f : X → Y be a function in associated w-spaces.Then f is said to be: (1) WO-continuous [10] if for x ∈ X and for each open set V containing f (x), there is a w-open set U containing x such that f (U ) ⊆ V ; (2) WK-continuous [10] (3) gw τ -continuous [12] if for x ∈ X and for each open set (5) gw-continuous [16] if for x ∈ X and for each open set V containing f (x), there is a gw-open set U containing x such that f (U ) ⊆ V : Obviously, the following things are obtained: Theorem 3.4.In associated w-spaces, (1) every W O-continuous function is wg-continuous; (2) every W K-continuous function is wg * -continuous; (3) every gw τ -continuous function is wg-continuous;  The following example supports that the converses of the above theorem are not true in general.
Theorem 3.11.Let f : X → Y be a function in associated w-spaces.Then if f is wg-continuous, then for an m-family H wg-converging to x ∈ X, a filter < f (H) > converges to f (x).
Proof.Suppose that f is wg-continuous and H is an m-family wg-converging to x ∈ X.Since f is wg-continuous, for an open set V containing f (x), there exists a wg-open set U containing x such that f Theorem 3.12.Let f : X → Y be a bijective function in associated w-spaces.Then f is wg * -continuous iff for an m-family H wg-converging to x ∈ X, the filter < f (H) > converges to f (x).
Proof.Suppose that f is wg * -continuous and H is an m-family wg-converging to x ∈ X.Since f is wg * -continuous and surjective, it is satisfied that O We recall that: Let f : X → Y be a function on w-spaces.Then f is said to be W-continuous [9] if for x ∈ X and for each w-open set V containing f (x), there is a w-open set U containing x such that f (U ) ⊆ V .Theorem 3.13 ([9]).Let f : (X, w X ) → (Y, w Y ) be a function in two w-spaces.Then f is W -continuous if and only if f −1 (wI(B)) ⊆ wI(f −1 (B)) for B ⊆ Y .Theorem 3.14 ([15]).Let (X, w τ ) be an associated w-space with a topology τ and A ⊆ X.Then A is wg-open if and only if F ⊆ wI(A) whenever F ⊆ A and F is closed.

( 5 )
x ∈ wgI(A) if and only if there exists a w τ g-open set U containing x such that U ⊆ A. (6) x ∈ wgC(A) if and only if A ∩ V = ∅ for all wg-open set V containing x.

3 .
wg-continuity; wg * -continuity Definition 3.1.Let f : X → Y ) be a function in two associated w-spaces.Then f is said to be (1) wg-continuous if for x ∈ X and for each open set V containing f (x), there is a wg-open set U containing x such that f

( 5 )
and (6) Since every gw-open set is wg-open, it is obtained.
Consider a function f : (X, w) → (X, w) defined by f (a) = d; f (b) = b; f (c) = c; f (d) = a.Then obviously f is wg-continuous and wg * -continuous.But for a open set V = {a, b}, since f −1 (V ) = {b, d} is neither gw-open nor gw τ , f is neither gw * -continuous nor gw * τ -continuous.Obviously, it implies that f is neither gw-continuous nor gw τ -continuous.Furthermore, since for a open set V = {a, b}, f −1 (V ) is not w-open, f is neither W K-continuous nor W O-continuous.Remark 3.6.For a function from an associated w-space to an associated w-space, we have the following diagram:Continuity ⇒ W K-continuity ⇒ W O-continuity ⇓ ⇓ gw * -continuity ⇒ gw-continuity ⇓ ⇓ wg * -continuity ⇒ wg-continuity ⇑ ⇑ gw * τ -continuity ⇒ gw τ -continuityTheorem 3.7.Let f : X → Y be a function in associated w-spaces.Then the following statements are equivalent:

Theorem 3 . 15 .
Let f : X → Y be a function in associated w-spaces.Then if f is W -continuous and closed, then f is wg * -continuous, i.e., for every open subset B in Y , f −1 (B) is wg-open.Proof.Let B be any open subset in Y and F be a closed set in X such thatF ⊆ f −1 (B).Now, we show that F ⊆ wI(f −1 (B)).Since B is open and f (F ) is closed, f (F ) ⊆ B = int(B) ⊆ wI(B).From (4) of Theorem 3.13, it follows that F ⊆ f −1 (wI(B)) ⊆ wI(f −1 (B)).Hence, f −1 (B) is wg-open.Hence, f is wg * -continuous.