w-SEMIOPEN SETS AND w-SEMICONTINUITY IN WEAK SPACES

The purpose of this short note is to generalize w-open sets in w-spaces, say w-semiopen sets. First, we introduce the notion of w-semiopen sets and some basic properties of such the notion. Second, we introduce W -semicontinuous functions defined by w-semiopen sets and investigate characterizations of them by using the several types of operators. AMS Subject Classification: 54A05, 54B10, 54C10


Introduction
Siwiec [16] 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 [13].The weak neighborhood system induces a weak neighborhood space which is independent of neighborhood spaces [4] and general topological spaces [2].The notions of weak structure, wspace, W -continuity and W * -continuity were investigated in [14].In fact, the set of all g-closed subsets [5] in a topological space is a kind of weak structure.
The one purpose of our research is to generalize w-open sets in w-spaces.In 1963, Levine [6] introduced the concept of semi-open set in topological spaces and used this to define other new concepts such as semi-closed set and semicontinuity of a function.In this paper, in the same way as Levine did, we are going to study the notion of w-semiopen set in a weak space.
First, we investigate the notion of w-semiopen sets and some basic properties of such the notion.Second, we introduce W -semicontinuous functions defined by w-semiopen sets, and study characterizations of them by using the operators wC, wI, wsC and wsI.

Preliminaries Definition 1 ([14]
).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 . ( The collection of all w-open sets (resp., w-closed sets) in a w-space X will be denoted by W O(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 [10] (resp., α-open set [15], semi-open [6]) 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.
A subset A of a topological space (X, τ ) is said to be: Then the family τ , GO(X), gαO(X), gα * O(X), gα * * O(X), αgO(X) and α * * gO(X) are all 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 [10] if ∅ ∈ w X and X ∈ w X .Thus clearly every weak structure is a minimal structure.

Definition 2 ([14]
).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) Theorem 3 ([14]).Let (X, w X ) be a w-space and A ⊆ X.
(1) x ∈ wI(A) if and only if there exists an element U ∈ W (x) such that U ⊆ A. ( , w-open), then wC(A) = A (resp., wI(A) = A).

Definition 4 ([14]
).Let f : X → Y be a function on w-spaces (X, w X ) and (Y, w Y ).Then f is said to be W -continuous [14] if for x ∈ X and V ∈ W (f (x)), there is U ∈ W (x) such that f (U ) ⊆ V .
3. w-Semiopen Sets, W -Semicontinuity Definition 5. Let (X, w X ) be a w-space and S ⊆ X.Then S is called a w-semiopen set if S ⊆ wC(wI(S)).The complement of a w-semiopen set is called a w-semiclosed set.
The family of all w-semiopen sets in X will be denoted by W SO(X). Lemma 6.Let (X, w X ) be a w-space and A ⊆ X.Then the following things hold: (1) A is a w-semiclosed set if and only if wI(wC(A)) ⊆ A.
We can easily show that a w-semiopen set may not be w-open as the next example: Example 7. Let X = {a, b, c} and w X = {∅, {a}, {b}, X} be a w-structure in X.Consider A = {a, b}; then wC(wI(A)) = X.So A is w-semiopen but not w-open.Theorem 8. Let (X, w X ) be a w-space.Any union of w-semiopen sets is also w-semiopen.
Proof.Let A i be a w-semiopen set for i ∈ J. From (3) of Theorem 2.3, A i ⊆ wC(wI(A i )) ⊆ wC(wI(∪A i )).Clearly, this implies ∪A i ⊆ wC(wI(∪A i )) and so ∪A i is w-semiopen.
In general, the intersection of any two w-semiopen sets is not w-semiopen.See the next example: Example 9.In example 3.3, consider A = {a, c} and B = {b, c}.Note that wC(wI(A)) = A and wC(wI(B)) = B.So both A and B are w-semiopen, but A ∩ B = {c} is not w-semiopen.Definition 10.Let (X, w X ) be a w-space and A ⊆ X.Then the w-semiclosure of A and the w-semi-interior of A, denoted by wsC(A) and wsI(A), respectively, are defined as: Theorem 11.Let (X, w X ) be a w-space and A, B, F ⊆ X.Then the following things hold: (1) wsI(A) ⊆ A; A ⊆ wsC(A).
Theorem 12. Let (X, w X ) be a w-space and A ⊆ X.Then (1) x ∈ wsC(A) if and only if A ∩ V = ∅ for every w-semiopen set V containing x.
(2) x ∈ wsI(A) if and only if there exists a w-semiopen set U such that U ⊆ A.
Proof.(1) Suppose that there is a w-semiopen set V containing x such that ∈ X − V .This implies x / ∈ wsC(A).The reverse relation is obvious.
(2) Obvious.Now, we introduce the notion of W -semicontinuous functions defined by w-semiopen sets, and study characterizations of them by using the operators wC, wI, wsC and wsI.Definition 13.Let f : (X, w X ) → (Y, w Y ) be a function on w-spaces X and Y .Then f is said to be W -semicontinuous if for each x and each w-open set V containing f (x), there exists a w-semiopen set U containing x such that f (U ) ⊆ V .
Since every w-open set is w-semiopen, obviously, a W -continuous function is W -semicontinuous, but the converse is not true in general.
Let f : (X, w 1 ) → (X, w 2 ) be the identity function.Then f is W -semicontinuous but not W -continuous.