DECOMPOSITION OF WEAK CONTINUOUS FUNCTIONS ... - ijpam

International Journal of Pure and Applied Mathematics Volume 117 No. 4 2017, 631-644 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: 10.12732/ijpam.v117i4.8

AP ijpam.eu

DECOMPOSITION OF WEAK CONTINUOUS FUNCTIONS Ennis Rosas1 , Carlos Carpintero2 , John Moreno3 , Jos´e Sanabria4 § 1 Departamento

de Ciencias Naturales y Exactas Universidad de la Costa Barranquilla, COLOMBIA 2 Vicerrector´ ıa de Investigaci´on Universidad Aut´onoma del Caribe Barranquilla, COLOMBIA 3,4 Facultad de Ciencias B´ asicas Universidad del Atl´antico Barranquilla, COLOMBIA 1, 2, 4 Departamento de Matem´ aticas Universidad de Oriente Cuman´ a, VENEZUELA

Abstract: Using the notion of w-space on X, we introduce the concepts of locally w-regular closed, locally w-regular semi closed, locally w-semi closed as a generalization of locally wclosed sets, its relationship between them are given and a new weak decomposition of some type of weak continuous functions are studied and characterized. AMS Subject Classification: 54A05, 54B08, 54D30 Key Words: locally w-sclosed, locally w-rsclosed, w-st-set, b w-sB-set, b weak (w, σ)-srcontinuous functions

1. Introduction In the last years, different forms of open sets are being studied. Recently, a sigReceived:

June 2, 2017

Revised:

October 26, 2017

Published:

January 25, 2018

§ Correspondence author

c 2017 Academic Publications, Ltd.

url: www.acadpubl.eu

632

E. Rosas, C. Carpintero, J. Moreno, J. Sanabria

nificant contribution to the theory of generalized open sets have been presented by A. Cs´asz´ ar [1], [2], [3]. Specifically, in 2002, A. Cs´asz´ar [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 [9], 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 W. K. Min in 2017 [6], introduced and studied the notions of weakly ωτ g-closed set and weakly ωτ g-open set as a generalization of the ωτ g-closed set and ωτ g-open set in associated w-spaces. In this article, using the notion of w-regular open set and w-regular semi open set, we introduce the concept of locally w-semi closed sets and locally w-semi regular semi closed and give a new weak theory of 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.

2. Preliminaries Definition 2.1. [7] Let X be a nonempty set. A subfamily wX of the power set P (X) is called a weak structure on X if it satisfies the following : 1. ∅ ∈ wX and X ∈ wX . 2. For U1 , U2 ∈ wX , U1 ∩ U2 ∈ wX Then the pair (X, wX ) is called a w-space on X. An element U ∈ wX is called w-open set and the complement of a w-open set is a w-closed set Definition 2.2. called:

Let (X, τ ) be a topological space. A subset A of X is

1. Semi open if there exists an open set U such that U ⊆ A ⊆ cl(U ). 2. Regular open if A = int(cl(A)).

DECOMPOSITION OF WEAK CONTINUOUS FUNCTIONS

633

3. Regular semi open if there exists a regular open set V such that V ⊆ A ⊆ cl(V ). The collection of all semi open sets is denoted by SO(X), the collection of all regular open sets is denoted by RO(X) and the collection of all regular semi open sets is denoted by RSO(X). Definition 2.3. [7] Let (X, wX ) be a w-space. For a subset A of X, T 1. The w-closure of A is defined as wC(A) = {F : A ⊆ F, X \ F ∈ wX }. S 2. The w-interior of A is defined as wI(A) = {U : U ⊆ A, U ∈ wX }. Theorem 2.1. [7] Let (X, wX ) 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) 3. wC(X \ A) = X \ wI(A) and wI(X \ A) = X \ wC(A) 4. If A is w-closed (resp.w-open), then wC(A) = A(resp.wI(A) = A) Definition 2.4. [8] Let (X, wX ) be a ω-space on X. A subset A of X is called w-semi open if A ⊆ wC(wI(A)). The complement of a w-semi open set is called w-semi closed. The collection of all w-semi open sets is denoted by wSO(X, wX ) and the collection of all w-semi closed sets is denoted by wSC(X, ωX ) Definition 2.5. [8] Let (X, wX ) be a w-space on X. For a subset A of X T 1. The w-semi closure of A is defined as wsC(A) = {F : A ⊆ F, X \ F ∈ wSO(X, wX )}. S 2. The w-semi interior of A is defined as wsI(A) = {U : U ⊆ A, U ∈ wSO(X, wX )}. Theorem 2.2. [8] Let (X, wX ) be a w-space on X. A, B subsets of X. Then the following hold: 1. wsI(A) ⊆ A and A ⊆ wsC(A). 2. If A ⊆ B, then wsI(A) ⊆ wsI(B) and wsC(A) ⊆ wsC(B). 2. wsI(wsI(A)) = wsI(A) and wsC(wsC(A)) = wsC(A).

634

E. Rosas, C. Carpintero, J. Moreno, J. Sanabria

3. wsC(X \ A) = X \ wsI(A) and wsI(X \ A) = X \ wsC(A). 4. A is w-semi closed (resp. w-semi open), if and only if wsC(A) = A (resp. wsI(A) = A). Theorem 2.3. [8] Let (X, wX ) be a w-space on X and A a subset of X. Then wsC(A) is a w-semi closed. Theorem 2.4. Let (X, wX ) 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. Example 2.1. Let X = N be the set of natural numbers. Define wX = {∅, {1}, N } ∪ P ({2n : n ∈ N }). The set of w-closed sets={∅, N, N \ {1}} ∪ {Ac : A ∈ P ({2n : n ∈ N })}. The set of w-semiopen sets={∅, {1}, N, F1 , F2 } ∪ P ({2n : n ∈ N }) where F1 ∩ {2n : n ∈ N } = 6 ∅ and 1 ∈ F2 . If we take A = {3}, wsC(A) = {3}, wC(A) = {2n + 1 : n ∈ N } and wI(wC(A)) = {1}. Observe that A ∪ wI(wC(A)) = {1, 3} ⊃ {3} = wsC(A). Theorem 2.5. Then

Let (X, wX ) be a w-space on X and A, B subsets of X.

1. x ∈ wsC(A) if and only if A ∩ V 6= ∅, for every w-semiopen set V containing x. 2. wsC(A ∩ B) = wsC(A) ∩ wsC(B).

3. Locally w-Semi Regular Semi Closed Sets Throughout this paper (X, wX , τ ) a weak topological space denotes (X, wX ) a w-space and (X, τ ) a topological space. Definition 3.1. Let (X, wX , τ ) be a weak topological space. A subset A of X is called locally w-regular closed if A = U ∩ F where U ∈ RO(X) and F is w-closed. Remark 3.1. If (X, wX , τ ) is a weak topological space, then 1. Every regular open set as well as a w-closed set is locally w-regular closed. 2. A ⊆ X is locally w-regular closed if and only if X − A is the union of a regular closed and a w-open set.

DECOMPOSITION OF WEAK CONTINUOUS FUNCTIONS

635

Example 3.1. Let X = {a, b, c} with topology τ = {∅, X, {a}, {b}, {a, b}} and weak structure wX = {∅, X, {a}, {b}}. 1. The set of regular open sets={∅, X, {a}, {b}}. 2. The set of locally w-regular closed = {∅, X, {a}, {b}, {a, c}, {b, c}}. Definition 3.2. Let (X, wX , 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. Every locally w-regular closed is locally w-closed, but the converse is not necessarily true. Example 3.2. Using Example 3.1, the set {a, b} is locally w-closed but is not locally w-regular closed. Definition 3.3. Let (X, wX , τ ) be a weak topological space. A subset A of X is called locally w-regular semi closed (briefly locally w-rsclosed ) if A = U ∩ F where U ∈ RO(X) and F is w-semi closed. Remark 3.3. If (X, wX , τ ) is a weak topological space, then 1. Every regular open set as well as a w-closed set is locally w-regular semi closed. 2. A is locally w-regular semi closed if and only if X − A is the union of a regular closed and a w-semi open set. Example 3.3. Using Example 3.1, we obtain: 1. The set of w-semi open sets={∅, X, {a}, {b}, {a, b}, {b, c}, {a, c}}. 2. The set of w-semi closed sets={∅, X, {a}, {b}, {c}, {b, c}, {a, c}}. 3. The set of locally w-rsclosed = {∅, X, {a}, {b}, {c}, {a, c}, {b, c}}. Definition 3.4. Let (X, wX , τ ) be a weak topological space. A subset A of X is called locally w-semi closed (briefly locally w-sclosed) if A = U ∩ F where U ∈ τ and F is w-semi closed. Remark 3.4. If (X, wX , τ ) is a weak topological space, then: 1. Every locally w-regular semi closed is locally w-semi closed, but the converse is not necessarily true. 2. Every locally w-regular closed is locally w-semi closed, but the converse is not necessarily true.

636

E. Rosas, C. Carpintero, J. Moreno, J. Sanabria

Example 3.4. Using Example 3.1, we obtain: 1. The set of regular open sets={∅, X, {a}, {b}}. 2. The set of w-semi closed sets={∅, X, {a}, {b}, {c}, {b, c}, {a, c}}. 3. The set of locally w-regular closed = {∅, X, {a}, {b}, {a, c}, {b, c}}. 4. The set of locally w-closed = {∅, X, {a}, {b}, {a, b}, {a, c}, {b, c}}. 5. The set of locally w-rsclosed = {∅, X, {a}, {b}, {c}, {a, c}, {b, c}}. 6. The set of locally w-sclosed = P(X). Theorem 3.1. Let (X, wX , τ ) be a weak topological space. A ⊆ X is locally w-regular semi closed if and only if there exists a regular open set U such that A = U ∩ wsC(A). Proof. Let A be a locally w-regular semi closed subset of X, then A = U ∩F , where U ∈ RO(X) and F is w-semi closed. Follows that A = A ∩ U ⊆ U ∩ wsC(A) ⊆ U ∩ wsC(F ) = U ∩ F = A. In consequence, A = U ∩ wsC(A). Conversely. Since wsC(A) is a w-semi closed. Follows that A is locally w-semi closed. Theorem 3.2. Let (X, wX , τ ) be a weak topological space. If A ⊆ B ⊆ X and B is locally w-regular semi closed, then there exists a locally w-regular semi closed C such that A ⊆ C ⊆ B. Proof. Suppose that B is locally w-regular semi closed, by Theorem 3.1, B = U ∩ wsC(B), where U is a regular open. Follows A ⊆ B ⊆ U , in consequence, A ⊆ U ∩ wsC(B), thus A ⊆ U ∩ wsC(A). If we take C = U ∩ wsC(A), C is locally w-regular semi closed and A ⊆ C ⊆ B. We know that if (X, wX , τ ) is a weak topological space, if A ⊆ X, wsC(A) is w-semi closed and then locally w-semi closed. In the case that A is locally w-regular semi closed, there exists some additional condition in order to obtain that wsC(A) \ A is locally w-regular semi closed. Example 3.5. Let X = {a, b, c} with topology τ = {∅, X, {a}, {b}, {a, b}} and weak structure wX = {∅, X, {b, c}}. Observe that the set A = {b} is locally w-regular semi closed, wsC(A) = X and wsC(A) − A = {b, c} is not locally w-regular semi closed. In the same form, A ∪ (X − wsC(A)) = A = {b} is not a w-semi open set. Also A is not contained in wsI(A∪(X −wsC(A))), because A ∪ (X − wsC(A)) = {a} and wsI({a}) = ∅.

DECOMPOSITION OF WEAK CONTINUOUS FUNCTIONS

637

In the case that RO(X) ⊂ wX , we have the following theorem. Theorem 3.3. Let (X, wX , τ ) be a weak topological space such that RO(X) ⊂ wX . If A is locally w-regular semi closed. Then: 1. wsC(A) − A is w-semi closed. 2. A ∪ (X − wsC(A)) is w-semi open set. 3. A is contained in wsI(A ∪ (X − wsC(A))). Proof. 1.-Suppose that A is a locally w-regular semi closed subset of X, then there exists a regular open set U such that A = U ∩ wsC(A). Follows that: wsC(A) − A = wsC(A) ∩ ((X − U ) ∪ (X − wsC(A))) = wsC(A) ∩ (X − U ) ∪ wsC(A) ∩ (X − wsC(A)) = wsC(A) ∩ (X − U ). Now wsC(A) is w-semi closed and X − U is regular closed and RO(X) ⊂ wX , we obtain that X − U is w-closed and then, X − U is w-semi closed. It follows that wsC(A) ∩ (X − U ) is w-semi closed. 2.- Using (1), wsC(A) − A is w-semi closed, then its complement X − (wsC(A)−A) is w-semi open, but X −(wsC(A)−A) = X −(wsC(A)∩(X −A) = A ∪ (X − wsC(A)). 3.-Using (2), A ⊂ (A ∪ (X − wsC(A))) = wsI(A ∪ (X − wsC(A))). Definition 3.5. Let (X, wX , τ ) be a weak topological space. A subset A of X is called locally w-semi regular semi closed (briefly locally w-srsclosed ) if A = U ∩ F where U ∈ RSO(X) and F is w-semi closed. Remark 3.5. If (X, wX , τ ) is a weak topological space, then: 1. Every regular semi open set as well as a w-semi closed set is locally w-semi regular semi closed. 2. A is locally w-semi regular semi closed if and only if X − A is the union of a regular semi closed and a w-semi open set. Remark 3.6. Every locally w-regular semi closed is locally w-semi regular semi closed, but the converse is not necessarily true. Example 3.6. Using Example 3.1, we obtain {c} is a locally w-semi regular semi closed set but is not regular semi open. Example 3.7. Using Example 3.5, we obtain {a, c} is a locally w-srsclosed set but is not locally w-regular sclosed, {b} is a locally w-semi regular semi closed set but is not w-semi closed.

638

E. Rosas, C. Carpintero, J. Moreno, J. Sanabria

Theorem 3.4. Let (X, wX , τ ) be a weak topological space. A ⊆ X is locally w-semi regular semi closed if and only if there exists a regular semi open set U such that A = U ∩ wsC(A). Proof. Let A be a locally w-semi regular semi closed subset of X, then A = U ∩ F , where U ∈ SRO(X) and F is w-semi closed. Follows that A = A ∩ U ⊆ U ∩ wsC(A) ⊆ U ∩ wsC(F ) = U ∩ F = A. In consequence, A = U ∩ wsC(A). Conversely, since wsC(A) is a w-semi closed. Follows that A is locally w-semi regular semi closed. Remark 3.7. Every locally w-regular semi closed is locally w-semi closed but the converse is false as shown by the next example. Theorem 3.5. Let (X, wX , τ ) be a weak topological space. If A ⊆ B ⊆ X and B is locally w-semi regular semi closed, then there exists a locally w-semi regular semi closed C such that A ⊆ C ⊆ B. Proof. Suppose that B is a locally w-semi regular semi closed, by Theorem 3.4, B = U ∩ wsC(B), where U is a regular semi open. Follows A ⊆ B ⊆ U , in consequence, A ⊆ U ∩ wsC(B), Thus A ⊆ U ∩ wsC(A). If we take C = U ∩ wsC(A), C is locally w-semi regular semi closed and A ⊆ C ⊆ B. It is possible to find some additional condition in order to prove that if A is locally w-semi regular semi closed, under what conditions wsC(A)−A is locally w-semi regular semi closed. Example 3.8. Using Example 3.5, RSO(X) = {∅, X, {a}, {b}, {a, c}, {b, c}}. If we take A = {a, c}, scw (A) = X and scw (A) − A = {b} is not w-semi closed. In the case that RSO(X) ⊂ wX , we have the following theorem. Theorem 3.6. Let (X, wX , τ ) be a weak topological space, such that RSO(X) ⊂ wX . If A is locally w-semi regular semi closed then: 1. wsC(A) − A is w-semi closed. 2. A ∪ (X − wsC(A)) is w-semi open set. 3. A is contained in wsI(A ∪ (X − wsC(A))). Proof. 1.-Suppose that A is a locally w-semi regular semi closed subset of X, then there exists a regular semi open set U such that A = U ∩ wsC(A). Follows that: wsC(A) − A = wsC(A) − (U ∩ wsC(A)) = wsC(A) ∩ (X − U ) ∪ wsC(A) ∩ (X − wsC(A)) = wsC(A) ∩ (X − U ). Now wsC(A) is w-semi closed

DECOMPOSITION OF WEAK CONTINUOUS FUNCTIONS

639

and X − U is semi regular closed and RSO(X) ⊂ wX , we obtain that X − U is w-semi closed and then wsC(A) ∩ (X − U ) is w-semi closed. 2.- Using (1), wsC(A) − A is w-semi closed, then its complement X − (wsC(A)−A) is w-semi open, but X −(wsC(A)−A) = X −(wsC(A)∩(X −A) = A ∪ (X − wsC(A)). 3.-Using (2), A ⊂ (A ∪ (X − wsC(A))) = wsI(A ∪ (X − wsC(A))). At this point there are a question, there exists any relation between the locally w-semi closed set and the locally w-semi regular semi closed set. We know that between the open sets and the regular semi open sets there are no relations, in this case, we affirm that both concepts are independent as is shown in the next example. Example 3.9. Using Example 3.5, we obtain that {a, b} is locally w-semi closed set but is not locally w-srsclosed set in the same form {a, c} is a locally w-srsclosed set but is not locally semi w-semi closed. Definition 3.6. Let (X, wX , τ ) be a weak topological space. A subset A of X is called: 1. w-t-set b if int(cl(A)) = int(cl(wC(A))). 2. w-B-set b if A = U ∩ V , U ∈ RO(X), V is a ω b -t-set. 3. w,, -open set if A ⊆ int(cl(wC(A))). Remark 3.8. If (X, wX , τ ) is a weak topological space, then 1. If A is a w-closed set then it is a w-t-set. b 2. If A is a w-t-set b then it is a w-B-set. b 3. Every locally w-regular closed set is a w-B-set. b Example 3.10. Let X = {a, b, c, d} with topology τ = {∅, X, {a}, {b}, {a, b}}. Consider the weak structure wX = {∅, X, {a}, {a, b}, {a, b, c}}. Observe that: {c} is a w-t-set b but is not a w-closed set. {a} is a w-B-set b but is not a w-t-set. b {a, b, c} is a µ b-B-set but is not locally w-regular closed.

Definition 3.7. Let (X, wX , τ ) be a weak topological space. A subset A of X is called: 1. w-st-set b if int(cl(A)) = int(cl(wsC(A))). 2. w-sB-set b if A = U ∩ V , U ∈ RO(X), V is a ω b -st-set.

640

E. Rosas, C. Carpintero, J. Moreno, J. Sanabria

3. w,, -sopen set if A ⊆ int(cl(wsC(A))). c 4. wb sB-set if A = U ∩ V , U ∈ SRO(X), V is a w-st-set. b Theorem 3.7. Let (X, wX , τ ) be a weak topological space, then: 1. If A is a w-semiclosed set then it is a w-st-set. b 2. If A is a w-st-set b then it is a w-sB-set. b 3. Every locally w-regular sclosed set is a w-sB-set. b c 4. If A is a w-sB-set b then it is a wb sB-set. Proof. 1. Let A be a w-semi closed set, then A = wsC(A). It follows that int(cl(A)) = int(cl(wsC(A))), in consequence, A is w-st-set. b 2. Suppose that A is a w-st-set. b Since X is a regular open set and A = A∩X, A is a w-sB-set. b 3. It follows from Definition 3.3 and 2. 4. Since every regular open set is regular semi open, the result follows. The following examples shows that the converse of Theorem 3.7 not necessarily is true. Example 3.11. Using Example 3.5, we obtain that: {a, b} is w-st-set b but is not w-semi closed. c {a, c} is wb sB-set but is not w-st-set. b {a, b} is w-sB-set b but is not locally w-regular semi closed. {b} is a w-sB-set b but is not w-st-set. b 4. Generalized w-Semiregular Semiclosed Sets Definition 4.1. Let (X, wX , τ ) be a weak topological space. A subset A of X is called generalized w-semi closed (briefly gw-sclosed) if wsC(A) ⊆ U where A ⊆ U and U ∈ τ . Definition 4.2. Let (X, wX , τ ) be a weak topological space. A subset A of X is called generalized w-regular closed (briefly gw-rclosed) if wC(A) ⊆ U where A ⊆ U and U ∈ RO(X). Definition 4.3. Let (X, wX , τ ) be a weak topological space. A subset A of X is called generalized w-regular semiclosed (briefly gw-rsclosed) if wsC(A) ⊆ U where A ⊆ U and U ∈ RO(X).

DECOMPOSITION OF WEAK CONTINUOUS FUNCTIONS

641

Definition 4.4. Let (X, wX , τ ) be a weak topological space. A subset A of X is called generalized w-semiregular closed (briefly gw-srclosed) if wC(A) ⊆ U where A ⊆ U and U ∈ RSO(X). Definition 4.5. Let (X, wX , τ ) be a weak topological space. A subset A of X is called generalized w-semiregular semiclosed (briefly gw-srsclosed) if wsC(A) ⊆ U where A ⊆ U and U ∈ RSO(X). Remark 4.1. It is easy to see that: 1. gw-srclosed ⇒gw-sclosed⇒gw-rsclosed. 2. gw-srclosed ⇒gw-srsclosed⇒gw-rsclosed. 3. gw-srclosed ⇒gw-rclosed⇒gw-rsclosed. But the converse are not necessarily true, as we can see in the following examples. Example 4.1. Using the Example 3.5, we obtain: {a, b} is a gw-rsclosed but is not gw-sclosed,{c}isagw−sclosed but is not gw-srclosed, {a, c} is a gwrsclosed but is not gw-srsclosed, {a, b} is a gw-rclosed but is not gw-srclosed, {a, b} is a gw-rclosed but is not gw-srclosed. Example 4.2. Using Example 3.6, we obtain that {b} is a gw-rsclosed but is not gw-rclosed. Theorem 4.1. Let (X, wX , τ ) be a weak topological space. A ⊂ X is w-closed if and only if A is gw-rclosed and locally w-rclosed. Proof. Suppose that A is w-closed and A ⊂ U where U ∈ RO(X). Since A = wC(A), we obtain that A is gw-rclosed and locally w-rclosed. Conversely. Suppose that A is gw-rclosed and locally w- rclosed, then A = U ∩ F , where U ∈ RO(X) and F is w-closed, therefore, A ⊂ U and A ⊂ F , in consequence, wC(A) ⊂ U and wC(A) ⊂ F . Follows that wC(A) ⊂ U ∩ F = A. So A is w-closed. Theorem 4.2. Let (X, wX , τ ) be a weak topological space. A ⊂ X is w-semiclosed if and only if A is gw-rsclosed and locally w-regular semiclosed. Proof. Suppose that A is w-semiclosed and A ⊂ U where U ∈ RO(X). Since A = wsC(A), we obtain that A is gw-rsclosed and locally w-srclosed. Conversely. Suppose that A is gw-rsclosed and locally w- srclosed, then A = U ∩ F , where U ∈ RO(X) and F is w-semiclosed, therefore, A ⊂ U and A ⊂ F , in consequence, wsC(A) ⊂ U and wsC(A) ⊂ F . Follows that wsC(A) ⊂ U ∩ F = A. So A is w-semiclosed.

642

E. Rosas, C. Carpintero, J. Moreno, J. Sanabria

Theorem 4.3. Let (X, wX , τ ) be a weak topological space. A ⊂ X is wsemiclosed if and only if A is gw-srsclosed and locally w-semi regular semiclosed. Proof. Suppose that A is w-semiclosed and A ⊂ U where U in RSO(X). Since A = wsC(A), we obtain that A is gw-srsclosed and locally w-srsclosed. Conversely. Suppose that A is gw-srsclosed and locally w- semiregular semiclosed, then A = U ∩ F , where U ∈ RSO(X) and F is w-semiclosed, therefore, A ⊂ U and A ⊂ F , in consequence, wsC(A) ⊂ U and wsC(A) ⊂ F . Follows that wsC(A) ⊂ U ∩ F = A. So A is w-semiclosed.

5. Weak Types of (w, σ)-Continuous Functions Definition 5.1. Let (X, wX , τ ) be a weak topological space and (Y, σ) be a topological space. A function f : (X, τ ) → (Y, σ) is weak (w, σ)-rcontinuous if f −1 (V ) is w-open in X for each regular open set V of Y. Definition 5.2. Let (X, wX , τ ) be a weak topological space and (Y, σ) be a topological space. A function f : (X, τ ) → (Y, σ) is weak (wX , σ)-rscontinuous if f −1 (V ) is w-semi open in X for each regular open set V of Y. Observe that every weak (wX , σ)-rcontinuous function is weak (wX , σ)rscontinuous function, but the converse is false. Example 5.1. Let X = Y = {a, b, c} with topology τ = {∅, X, {a}, {b}, {a, b}}, weak structure wX = {∅, X, {a}, {b}} and σ = {∅, X, {a}, {c}, {a, c}, {b, c}}. Define f : (X, τ ) → (Y, σ) as f (a) = c, f (b) = a and f (c) = b. Observe that: RO(Y ) = {∅, X, {a}, {b, c}}. The set of w-semi open sets={∅, X, {a}, {b}, {a, b}, {b, c}, {a, c}}, f −1 ({a}) = {b}, f −1 ({b, c}) = {a, c}. It follows that f is weak (wX , σ)-rscontinuous but is not weak (wX , σ)-rcontinuous. Definition 5.3. Let (X, wX , τ ) be a weak topological space and (Y, σ) be a topological space. A function f : (X, τ ) → (Y, σ) is weak (w, σ)-srscontinuous if f −1 (V ) is w-semi open in X for each regular semi open set V of Y. Observe that every weak (w, σ)-srscontinuous function is weak (w, σ)-rscontinuous function, but the converse is false. Example 5.2. Let X = Y = {a, b, c} with topology τ = {∅, X, {a}, {c}, {b, c}}, weak structure wX = {∅, X, {a}, {b}, {b, c}} and σ = {∅, Y, {a}, {b}, {a, b}}. Define f : (X, τ ) → (Y, σ) as f (a) = b, f (c) = a and f (b) = c. Observe that: RO(Y ) = {∅, X, {a}, {b}}, RSO(Y ) = {∅, X, {a}, {b}, {a, c}, {b, c}}, The set of w-semi open sets={∅, X, {a}, {c}, {a, b}, {b, c}, {a, c}}, f −1 ({a}) = {c},

DECOMPOSITION OF WEAK CONTINUOUS FUNCTIONS

643

f −1 ({b}) = {a}, f −1 ({c}) = {b}, f −1 ({a, c}) = {b, c}, f −1 ({b, c}) = {a, b} . It follows that f is weak (wX , σ)-rscontinuous but is not weak (wX , σ)srscontinuous. Definition 5.4. Let (X, wX , τ ) be a weak topological space and (Y, σ) be a topological space. Then f : (X, τ ) → (Y, σ) is said to be weak generalized w-regular semi continuous(briefly weak gw-rscontinuous) (respectively weak contra locally generalized w-regular semi continuous (briefly weak contra locally gw-rscontinuous)) if f −1 (F ) is a gw-rsclosed (respectively locally w-rsclosed) for each regular closed set F of (Y, σ). Example 5.3. Using Example 3.1. Define f : (X, τ ) → (X, τ ) as the identity function, the it is easy to proof the f is weak gw-rscontinuous and weak contra locally gw-rscontinuous. Definition 5.5. Let (X, wX , τ ) be a weak topological space and (Y, σ) be a topological space. Then f : (X, τ ) → (Y, σ) is said to be weak generalized w-semi regular semi closed (briefly weak gw-srscontinuous) (respectively weak contra locally generalized w-semi regular semi continuous(briefly weak contra locally gw-srscontinuous)) if f −1 (F ) is a gw-srsclosed (respectively locally wsrsclosed) for each regular semi closed set F of (Y, σ). Example 5.4. Using Example 3.1. Define f : (X, τ ) → (X, τ ) as the identity function, then it is easy to proof that f is weak gw-srscontinuous and weak contra locally gw-srscontinuous. Theorem 5.1. Let (X, wX , τ ) be a weak topological space and (Y, σ) be a topological space. Then for a function f : (X, τ ) → (Y, σ), the following are equivalent: 1. f is weak (w, σ)-rscontinuous. 2. f −1 (F ) is w-semi closed for each regular semi closed set F of (Y, σ). Proof. The proof is clear. Theorem 5.2. Let (X, wX , τ ) be a weak topological space and (Y, σ) be a topological space. Then for a function f : (X, τ ) → (Y, σ), the following are equivalent: 1. f is weak (w, σ)-srscontinuous. 2. f −1 (F ) is w-semi closed for each regular semi closed set F of (Y, σ). Proof. The proof is clear.

644

E. Rosas, C. Carpintero, J. Moreno, J. Sanabria

Theorem 5.3. Let (X, wX , τ ) be a weak topological space and (Y, σ) be a topological space. Then f : (X, τ ) → (Y, σ) is weak (w, σ)-rscontinuous if and only if it is weak gw-rscontinuous and weak contra locally gw-rscontinuous. Proof. The proof follows from Theorem 4.2 and Theorem 5.1. In analogous form, we can proof the following theorem. Theorem 5.4. Let (X, wX , τ ) be a weak topological space and (Y, σ) be a topological space. Then f : (X, τ ) → (Y, σ) is weak (w, σ)-rscontinuous if and only if it is weak gw-rscontinuous and weak contra locally w-rscontinuous. Proof. The proof follows from Theorem 4.3 and Theorem 5.2.

References [1] A. Cs´ aszar, Generalized topology, generalized continuity, Acta Math. Hungar., 96 (2002),351-357, doi: 10.1023/A:1019713018007. [2] A. Cs´ aszar, Generalized open sets in generalized topologies, Acta Math. Hungar., 96 (2005), 53-66, doi: 10.1007/s10474-005-0005-5. [3] A. Cs´ aszar, δ and θ-modifications of generalized topologies, Acta Math. Hungar., 120, No. 3 (2008), 275-279, doi: 10.1007/s10474-007-7136-9. [4] Bishwambhar Roy and Ritu Sen, On a type of decomposition of continuity, Afr. Mat., 26, No. 1-2 (2015), 153-158, doi: 10.1007/s13370-013-0195-x. [5] Bishwambhar Roy and Ritu Sen. On decomposition of weak continuity, Creat. Math. Inform., 24, No. 1-2 (2015), 83-88. [6] W.K. Min, On weakly wτ g-closed sets in associated w-spaces, International Journal of Pure an Applied Mathematics, 113, No. 1 (2017), 181-188, doi: 10.12732/ijpam.v112i2.18. [7] W.K. Min and Y.K. Kim, On weak structures and w-spaces, Far East Journal of Mathematical Sciences, 97, No. 5 (2015), 549-561. [8] W.K. Min and Y.K. Kim, w-semi open sets and w-semi continuity in weak spaces, International Journal of Pure an Applied Mathematics, 110, No. 1 (2016), 49-56, doi: 10.12732/ijpam.v110i1.5. [9] Carlos Carpintero, John Moreno and Ennis Rosas. Some New types of decomposition of continuity II, Creat. Math. Inform., 25 No. 1 (2016), 41-49.