MORE ON WEAK DECOMPOSITION OF CONTINUITY

Using the notion of w-space on a set X and the concept of locally w-semi open set, we introduce, study and characterize the notions of w-s-Kernel of a subset A of X. Also we introduce and study a new forms of weak decomposition of continuity. AMS Subject Classification: 54C05, 54C08, 54C10


Introduction and Preliminaries
In the last years, different variants of open sets are being studied.Recently, a significant contribution to the theory of generalized open sets have been presented by A. Császár [1], [2], [3].Specifically, in 2002, A. Császár [1], introduced the notions of generalized topology and generalized continuity.It is observed that a large numbers of articles are devoted to the study of generalized open sets and certain type of sets associated to a topological spaces, containing the class of open sets and possessing properties more or less to those open sets.Bishwambhar.et al. [4] studied some type of decomposition of continuity using generalized topologies and in [5], studied some weak forms of continuity.Rosas E. et al. in [10], give a new theory of decomposition of continuous functions using generalized topologies.In 2015, W. K. Min et al. [7], introduced and studied the notions of weak structures on a nonempty set X.In 2016, W. K. Min et al. introduced the notions of w-semiopen sets and w-semi continuity in w-spaces.Later in 2017, W. K. Min in [6], introduced and studied the notions of weakly w τ g-closed set and weakly w τ g-open set as a generalization of the w τ g-closed set and w τ g-open set in associated w-spaces.E. Rosas et al in [9], introduce the concepts of locally w-regular closed sets and locally w-semi regular semi closed and a new weak decomposition of some type of weak continuity functions are studied and characterized.In this article, using the notion of w-semi open set, we introduce the concept of locally w-semi open set as a generalization of locally w-closed and give a new theory of weak decomposition of continuity and some weak form of continuity are studied.Throughout this paper cl(A) (respectively int(A)) denotes the closure (respectively interior) of A in a topological space X.

Preliminaries
Definition 2.1.[7] 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 :

The w-interior of A is defined as wI
Theorem 2.3.[7] Let (X, w X ) be a w-space on X. A, B subsets of X.Then the following hold: 1.If A ⊆ B, then wI(A) ⊆ wI(B) and wC(A) ⊆ wC(B).
2. wI(wI(A)) = wI(A) and wC(wC(A)) = wC(A) The collection of all w-semi open sets is denoted by wSO(X, w X ) and the collection of all w-semi closed sets is denoted by wSC(X, w X ) Definition 2.5.[8] Let (X, w X ) be a w-space on X.For a subset A of X 2. The w-semi interior of A is defined as wsI(A) = {U : U ⊆ A, U ∈ wSO(X, w X )}.
Theorem 2.6.[8] Let (X, w X ) be a w-space on X. A, B subsets of X.Then the following hold: 1. wsI(A) ⊆ A and A ⊆ wsC(A).

wsI(wsI(A))
= wsI(A) and wsC(wsC(A)) = wsC(A).Theorem 2.7.[8] Let (X, w X ) be a w-space on X and A a subset of X.Then wsC(A) is an w-semi closed.

wsC(X
Theorem 2.8.Let (X, w X ) be a w-space on X and A a subset of X.Then A ∪ wI(wC(A)) ⊆ wsC(A).
The following example shows that the reverse contention in the above theorem is not necessarily true.

New Types of w-Closed Sets
Throughout this paper (X, w X , τ ) a weak topological space denotes (X, w X ) is a w-space and (X, τ ) is a topological space.
Definition 3.1.[4] Let (X, w X , τ ) be a weak topological space.A subset A of X is called locally w-closed if A = U ∩ F where U ∈ τ and F is w-closed.Remark 3.2.If (X, w X , τ ) is a weak topological space, then every open set as well as a w-closed set is locally w-closed.Theorem 3.3.Let (X, w X , τ ) be a weak topological space.If A ⊆ X is locally w-closed then there exists an open set U such that A = U ∩ wC(A).
Proof.Let A be a locally w-closed subset of X, then A = U ∩F , where U ∈ τ and Example 3.4.In Example 3.12, {a, b} = {a, b, c}∩wsC({a, b}), but {a, b} is not a locally w-closed set.Definition 3.5.Let (X, w X , τ ) be a weak topological space.A subset A of X is called: ) is a weak topological space, then every open set as well as a w-semi closed set is locally w-semi closed, also every locally w-closed set is locally w-semi closed.Definition 3.9.Let (X, w X , τ ) be a weak topological space.A subset A of X is called:

w-sB-set if
The following theorems characterizes the locally w-semi closed sets.
Theorem 3.11.Let (X, w X , τ ) be a weak topological space.A ⊆ X is locally w-semi closed if and only if there exists an open set U such that A = U ∩ wsC(A).
Proof.Let A be a locally w-semi closed subset of X, then A = U ∩F , where U ∈ τ and F is w-semi closed.It follows that In the following example, we can see that there exists a locally w-semi closed set that is not open as well as w-semi closed, w-st-set that is not w-s-set, w-sB-set that is not w-B-set.

If
A is a w-B-set, then A is a w-sB-set.
Example 3.15.Let X = {a, b, c} with τ = {∅, X, {a}, {b}, {a, b}} and weak structure w X = {∅, X, {b, c}} and on (X, τ ).Observe that τ is not contained in w.If we take A = {a, b}, A is locally w-semi closed, because In the case that τ ⊂ w X , we have the following theorem.Theorem 3.16.Let (X, w X , τ ) be a weak topological space and τ ⊂ w X .If A is locally w-semi closed, then: Proof.1.-Suppose that A is a locally w-semi closed subset of X, then there exists an open set U such that A = U ∩ wsC(A).It follows that: Definition 3.17.Let (X, w X , τ ) be a weak topological space.A subset A of X is called generalized w-closed or simple a gw-closed if wC(A) ⊆ U whenever A ⊆ U and U ∈ τ .Definition 3.18.Let (X, w X , τ ) be a weak topological space.A subset A of X is called generalized w-semi closed or simple a gw-s-closed if wsC(A) ⊆ U whenever A ⊆ U and U ∈ τ .Remark 3.19.In a weak topological space (X, w X , τ ), every generalized w-closed set is a generalized w-semi closed set, but the converse is not necessarily true as we can see in the following example.3. w-closed sets ={∅, X, {c, d}, {a, d}, {a, c, d}}.
The following theorems characterize: the w-closed sets in terms of gw-closed sets and locally w-closed sets and the w-semi closed sets in terms of gw-semi closed sets and locally w-semi closed sets.Theorem 3.21.Let (X, w X , τ ) be a weak topological space.A ⊂ X is w-closed if and only if A is gw-closed and locally w-closed.
Proof.Suppose that A is w-closed in X and A ⊂ U , with U ∈ τ .Since A = wC(A), we obtain that A is gw-closed and locally w-closed.Conversely, suppose that A is gw-closed and locally w-closed, then A = U ∩ F , where U ∈ τ and F is w-closed, therefore, A ⊂ U and A ⊂ F , in consequence, wC(A) ⊂ U and wC(A) ⊂ F and hence wC(A) ⊂ U ∩ F = A.So A is w-closed.Theorem 3.22.Let (X, w X , τ ) be a weak topological space.A ⊂ X is w-semi closed if and only if A is gw-semi closed and locally w-semi closed.
Proof.Suppose that A is w-semi closed in X and A ⊂ U , with U ∈ τ .Since A = wsC(A), we obtain that A is gw-semi closed and locally w-semi closed.Conversely, suppose that A is gw-semi closed and locally w-semi closed, then A = U ∩F , where U ∈ τ and F is w-semi closed, therefore, A ⊂ U and A ⊂ F , in consequence, wsC(A) ⊂ U and wsC(A) ⊂ F and hence wsC(A) ⊂ U ∩ F = A.So A is w-semi closed.Theorem 3.23.Let (X, w X , τ ) be a weak topological space and A, B subsets of X.

Every locally
In consequence, int(wsC(A)) = A, so A is a w-st-set.2-.If A is w-semi closed, then A = wsC(A), and hence int(A) = int(wsC(A)).Therefore, A is w-st-set.3-.Suppose that A and B are w-st-sets.2), F is w-st-set, then by ( 4), follows that A = U ∩ F , where U ∈ τ and F is a w-st-set and therefore, A is w-sB-set.
In the following examples show that the converse of the above theorem is not necessarily true.
Example 3.24.In Example 3.12, {c} is w-sB-set, but is not w-st-set.Also {a} is w-st-set, but is not w-semi closed.
Example 3.25.Let X = {a, b, c}, τ = {∅, X, {c}} and w X = {∅, X, {a}, {a, b}}.Then {a} is a w-sB-set but is not locally w-semi closed.Theorem 3.26.Let (X, w X , τ ) be a weak topological space.A ⊂ X is open if and only if A is w , -s-open and w-sB-set.
Example 3.27.In Example 3.12, {b, c, d} is w , -s-open but not w-sB-set.In the same form, {a} is w-sB-set but is not w , -s-open, in consequence, is not open.

2.
For each x ∈ X and each open set V of Y with f (x) ∈ V, there exists a w-semi open set U containing x such that f (U ) ⊆ V.

For each x ∈ X and each open set
4. The inverse image of each closed set in Y is w-semi closed.
2. Let A be w-semi open in X.Then always A ⊆ w-s-Ker(A).On the other hand, assume that x ∈ w-s-Ker(A).Then 3. It is obvious.
As immediate consequence of Theorems 4.5 and 4.11, we have the following result.

For every subset
Definition 4.13.Let (X, w X , τ ) be a weak topological space and (Y, σ) be a topological space.Then f : (X, τ ) → (Y, σ) is said to be gw-s-continuous (respectively contra locally w-s-continuous) if f −1 (F ) is a gw-s-closed (respectively locally w-s-closed) for each closed set F of (Y, σ).
Example 4.14.In Example 3.13, take f : (X, τ ) → (X, τ ), defined as: f (a) = b, f (b) = c and f (c) = a, then f is contra locally w-s-continuous but is not gw-s-continuous.In the same form if in Example 3.25, we define f : (X, τ ) → (X, τ ), as: f (a) = b, f (b) = a and f (c) = c, then f is w-scontinuous but is not contra locally w-s-continuous.Observe that in each case f is not (w, τ )-s-continuous.
The following theorem is a direct consequence of Theorem 4.5 and Theorem 3.22 Theorem 4.15.Let (X, w X , τ ) be a weak topological space and (Y, σ) be a topological space.Then f : (X, τ ) → (Y, σ) is (w, σ)-s-continuous if and only if it is gw-s-continuous and contra locally w-s-continuous.
The following example shows the existence of a function that is contra locally w-s-continuous but not is gw-s-continuous, in consequence is not (w, τ )s-continuous.

Example 4 .
16.In Example 3.12, define f : (X, τ ) → (X, τ ) as follows:f (a) = a, f (b) = d, f (c) = b and f (d) = c.According with 3.20, f is contra locally w-s-continuous but not is gw-s-continuous, in consequence is not (µ, τ )s-continuous.Theorem 4.17.Let (X, w X , τ ) be a weak topological space and (Y, σ) be a topological space.Then a contra continuous functionf : (X, τ ) → (Y, σ) is (w, σ)-s-continuous if and only if it is gw-s-continuous Proof.Suppose that f is contra continuous and (w, σ)-s-continuous.Let F be a closed set in Y , then f −1 (F ) is open and w-semi closed in X.Since each w-semi closed is gw-s-closed, then f is gw-s-continuous.Conversely, let F be a closed set in Y , then f −1 (F ) is open and gw-s-closed in X.Since each open set is locally w-s-closed, then f −1 (F ) is locally w-s-closed and gw-s-closed, by Theorem 3.22, f is (w, σ)-s-continuous.

Example 4 .
18.In Example 4.4, f is (µ, σ)-s-continuous, f (R \Q) = R \Q is w-open but is not w-s-closed in consequence, f is not contra continuous.