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On Soft Generalized Closed Sets in a Soft Topological Space with a Soft Weak Structure

Hanan S. Al-Saadi1, and Won Keun Min2

1Department of Mathematics, Faculty of Applied Sciences, Umm Al-Qura University, Makkah, Saudi Arabia, 2Department of Mathematics, Kangwon National University, Chuncheon, Korea
Correspondence to: Won Keun Min (wkmin@kangwon.ac.kr)
Received December 12, 2017; Revised December 20, 2017; Accepted December 27, 2017.
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract

In this work, we introduce the notion of generalized ω-closed set in a soft topological space with a soft weak structure. And some basic properties of this new class are investigated by using the concept of weak structure. Moreover, we study soft ω-T12-spaces defined by soft -closed sets and study some properties of it by using -open sets.

Keywords : Soft set, Soft topology, Soft weak structure, Soft generalized closed set
1. Introduction

As a solution to many problems, scientists have resorted to use approximate solution. In 1999, Molodtsove [1] initiated the concept of soft set theory as a new mathematical tool which is free from the problems mentioned above. Later on Maji et al [2] proposed several operations on soft sets and some basic properties and then Pei and Miao [3] investigated the relationships between soft sets and information systems.

In 2011, Shabir and Naz [4] introduced the notion of soft topological spaces and Min [5] corrected some their results. Zorlutuna et al. [6] continued to study the properties of soft topological spaces by defining the concepts of interior, soft neighborhoods in soft topological spaces. Varo and Aygun [7] presented soft Hausdorff spaces and introduced some new concepts such as convergence of sequences. Levine [8] introduced generalized closed sets in topological spaces.

In 2013, Cagman et al. [9] defined soft topological spaces by modifying the soft set. Also, Roy and Samanta [10] strengthen the definition of the soft topological spaces presented in [9] and they used the base and the subbase to characterize its properties.

In 1997, Csaszar [4] has studied generalized topological notions in collections which are closed under unions. Many other authors [1215] investigated the properties of the generalized topology. Recently, Császár [16] introduced a new notion called weak structures. Let X be a non-empty set and P(X) be its power set. A structure ω on X is called a weak structure (briefly, WS) on X if and only if φω [16].

In 2012, Al-Omari and Noiri [17] introduced and studied a kind of sets called generalized ω-closed (briefly, -closed) sets in topological space. In this paper, we introduce the notions of soft generalized ω-closed (briefly, soft -closed) sets and soft -open sets in soft topological spaces. Also, we will investigate some new soft separation axioms. In particular, we study soft ω-$T12$-spaces.

Let X be a non-empty set, E a set of parameters, P(X) denote the power set of X and A be a non-empty subset of E.

Definition 1.1 ( [1])

For AE, a pair (F, A) is called a soft set over X, where F is a mapping given by F: AP(X). For eA, F(e) may be considered as the set of e-approximate elements of the soft set (F, A).

Definition 1.2 ( [5])

A soft set (F, A) over X is said to be:

1. A null soft set denoted by ∅̃ if F(e) = ∅︀ for all eA.

2. An absolute soft set denoted by if F(e) = X for all eA.

Definition 1.3 ( [2, 5, 18])

For any two soft sets (F, A) and (G, B) defined over a common universe X, we have:

1. (F, A)⊂̃ (G, B) iff AB and F(e) ⊆ G(e) for all eA.

2. (F, A) ≅ (G, B) iff (F, A)⊂̃ (G, B) and (G, B)⊂̃ (F, A).

3. (F, A)⊂̃ (G, B) ≅ (H, C) where C = AB and

$H(e)={F(e),if e∈A-B,G(e),if e∈B-A,F(e)∪G(e),if e∈A∩B.$

for all eC.

4. (F, A)∩̃(G, B) ≅ (K, D) where D = AB and K(e) = F(e) ∩ G(e) for all eC.

5. X ∈ (F, A) where XX iff XF(e) for all eA and X ∉ (F, A) whenever XF(e) for some eA.

6. (F, E) ≃ (G, E) ≅ (M, E) where M(e) = F(e) − G(e) for all eE.

Definition 1.4 ( [19])

For a soft set (F, A) over X, the relative complement of (F, A) (denoted by (F, A)′) is defined by (F, A)′ ≅ (F′, A), where F′: AP(X) is given by F′(e) = XF(e) for all eA.

Definition 1.5 ( [4])

Let τ be the collection of soft sets over X, then τ is called a soft topology on X if τ satisfies the following axioms:

1. ∅̃, belong to τ.

2. The union of any number of soft sets in τ belong to τ.

3. The intersection of any two soft sets in τ belong to τ.

The triple (X, τ, E) is called a soft topological space over X. The member of τ are said to be soft open set in X. A soft set (F, E) over X is said to be soft closed in X if its relative complement (F, E)′ belong to τ.

Lemma 1.6 ( [14] )

Let (F, E) be a soft set over X and XX, then

1. X ∈ (F, E) iff (x, E)⊂̃(F, E)

2. if (x, E) ∩ (F, E) = ∅︀, then X ∉ (F, E).

Csaszar defined iω(A) as the union of all ω-open subsets of A and cω(A) as the intersection of all ω-closed sets containing A.

Let X be a non-empty set and E be a set of parameters. A collection ω of soft sets defined over X with respect to E is called a soft weak structure [20] iff ∅̃ ∈ ω. A soft set (F, A) is called ω-open soft set iff (F, A) ∈ ω and called ω-closed soft set iff (F, A)′ ∈ ω.

Definition 1.7 ( [19])

In a soft topological spaces (X, τ, E), a soft set (F, E) over X is called a soft generalized closed set (briefly, soft g-closed) if Cl(F, E) ⊆ (G, E) whenever (F, E) ⊆ (G, E) and (G, E) is soft open in (X, τ, E).

Lemma 1.8 ( [20])

Let ω be a soft weak structure defined over X with respect to E and (F, E), (G, E) are two soft sets over X, then:

1. iω(F, E)⊂̃(F, E)

2. If (F, E)⊂̃(G, E), then iω(F, E)⊂̃iω(G, E) and cω(F, E)⊂̃cω(G, E)

3. iω(iω(F, E)) ≅ iω(F, E) and cω(cω(F, E)) ≅ cω(F, E).

4. iω(F, E)′ = (cω(F, E))′ and cω(F, E)′ = (iω(F, E))′.

Lemma 1.9 ( [20])

Let ω be a soft weak structure defined over X with respect to the parameters set E and (F, E) be a soft set, then:

1. Xiω(F, E) if there exists an ω-open soft set (G, E) such that X ∈ (G, E)⊂̃(F, E).

2. Xcω(F, E) if and only if (G, E)∩̃(F, E) ≠ ∅̃ for all (G, E) ∈ ω such that X ∈ (G, E).

3. If (F, E) ∈ ω, then (F, E) = iω(F, E) and if (F, E) is ω-closed soft set, then (F, E) = cω(F, E)

Lemma 1.10 ( [4])

Let (X, τ, E) be a soft topological space over X. If (x, E) is a soft closed set in τ for each XX, then (X, τ, E) is a soft T1-space.

Definition 1.11 ( [20])

A soft topological space (X, τ, E) is called:

1. Soft ω-T0 if for each x, yX such that Xy, there exists a soft ω-open set (F, E) such that X ∈ (F, E) and y ∉ (F, E) or X ∉ (F, E) and y ∈ (F, E).

2. Soft ω-T1 if for each x, yX such that Xy, there exists soft ω-open sets (F, E) and (G, E) such that X ∈ (F, E), y ∉ (F, E), X ∉ (G, E) and y ∈ (G, E).

2. Soft -Closed Set

Definition 2.1

Let X be a non-empty set and E be a set of parameters. Let a collection ω be a soft weak structure on a soft topological space (X, τ, E). Then a soft set (F, E) over X is called a soft generalized ω-closed set (briefly, soft -closed set) if cω(F, E)⊂̃(G, E) whenever (F, E)⊂̃(G, E) and (G, E) is soft open in (X, τ, E). The complement of a soft -closed set is called a soft generalized ω-open (briefly, soft -open) set.

Example 2.2

Let X = {h1, h2, h3}, E = {e1, e2} and τ = {∅̃, X̃, (F1, E), (F2, E), (F3, E), (F4, E)} where

$F1(e1)={h2},F1(e2)={h1};F2(e1)={h2,h3},F2(e2)={h1,h2};F3(e1)={h1,h2},F3(e2)={h1,h3};F4(e1)={h1},F4(e2)={h3}.$

Let ω = {∅̃, (F1, E), (F3, E)} be a soft weak structure over X with respect to E. Then (G, E) is a soft -closed defined by G(e1) = ∅̃; G(e2) = {h2}, but (F1, E) is not soft -closed.

Remark 2.3

Let ω be a soft weak structure on a soft topological space (X, τ, E). Then every soft -closed set is soft g-closed. The following example shows that the converse need not be true in general.

Example 2.4

In Example 2.2, (F2, E) is soft g-closed in (X, τ, E) but not soft -closed over X.

Remark 2.5

For a soft weak structure ω on a soft topological space over X, every ω-closed set is a soft -closed set. In fact, if (F, E) is a ω-closed set (F, E)⊂̃(G, E) and (G, E) is soft open, then (F, E) = cω(F, E)⊂̃(G, E), so that (F, E) is soft -closed. The following example shows that the converse need not be true in general.

Example 2.6

In Example 2.2, (G, E) is soft -closed but not ω-closed.

Theorem 2.7

Let ω be a soft weak structure on a soft topological space (X, τ, E). If (F, E) is soft -closed in X and (F, E)⊂̃(H, E)⊂̃cω(F, E), then (H, E) is soft -closed

Proof

Suppose that (F, E) is soft -closed over X and (F, E)⊂̃(H, E)⊂̃cω(F, E). Let (H, E)⊂̃(G, E) and (G, E) is soft open in X. Since (F, E)⊂̃(H, E) and (H, E)⊂̃(G, E), we have (F, E)⊂̃(G, E). Since (F, E) is soft -closed, then cω(A, E)⊂̃(G, E). Since (H, E)⊂̃cω(F, E), we have cω(H, E)⊂̃cω(F, E)⊂̃(G, E). Therefore (H, E) is soft -closed.

The next example shows that the intersection of two soft -closed sets is not in general soft -closed.

Example 2.8

Let X = {h1, h2, h3}, E = {e1, e2} and τ = {∅̃, X̃, (F1, E), (F2, E), (F3, E), (F4, E)} where

$F1(e1)={h1},F1(e2)={h2};F2(e1)={h3},F2(e2)={h1,h3};F3(e1)={h1,h2},F3(e2)={h2,h3};F4(e1)={h1,h3},F4(e2)=∅.$

Let ω = {∅̃, (F2, E)} be a soft weak structure over X with respect to E. Then (H, E) be a soft -closed defined by H(e1) = H(e2) = {h2}. Thus it can be easily checked that (F1, E)∩̃(H, E) is not a soft -closed set.

Theorem 2.9

Let ω be a soft weak structure on a soft topological space (X, τ, E). If (F, E) is a soft -closed set, then cω(F, E) − (F, E) does not contain any non-empty soft closed set.

Proof

Let (H, E) be a soft closed subset of X such that (H, E)⊂̃cω(F, E) − (F, E), where (F, E) is soft -closed. Since (H, E)′ is soft open,(F, E)⊂̃(H, E)′ and (F, E) is soft -closed, cω(F, E)⊂̃(H, E)′ and thus (H, E)⊂̃[cω(F, E)]′. Thus (H, E)⊂̃[cω(F, E)]′ ∩̃cω(F, E) = ∅̃ and hence (H, E) = ∅̃.

If cω(F, E) − (F, E) does not contain any non-empty soft closed subset of X, then (F, E) need not be soft -closed in general.

Example 2.10

In Example 2.8, let (G, E) be a soft set defined by G(e1) = ∅̃, G(e2) = {h2}. Then cω(G, E) − (G, E) = (K, E) does not contain any non empty soft closed set such that K(e1) = {h1, h2}, K(e2) = ∅︀, but (G, E) is not a soft -closed set.

Corollary 2.11

Let ω be a soft weak structure on a soft topological space (X, τ, E) and (F, E) be a soft -closed set. Then cω(F, E) = (F, E) if and only if cω(F, E) − (F, E) is soft closed.

Proof

Let (F, E) be a soft -closed set. If cω(F, E) = (F, E), then cω(F, E)−(F, E) = ∅︀, and cω(F, E)−(F, E) is a soft closed set.

Conversely, let cω(F, E) − (F, E) be a soft closed set. Since (F, E) is soft -closed, then by Theorem 2.9, cω(F, E) − (F, E) does not contain any non-empty soft closed set. Since cω(F, E) − (F, E) is a soft closed subset of itself, cω(F, E) − (F, E) = ∅︀ and hence cω(F, E) = (F, E).

Theorem 2.12

Let ω be a soft weak structure on a soft topological space (X, τ, E). Then (H, E) is soft -open if and only if (F, E)⊂̃iω(H, E) whenever (F, E)⊂̃(H, E) and (F, E) is soft closed.

Proof

Let (H, E) be a soft -open set and (F, E)⊂̃(H, E), where (F, E) is soft closed. Then (H, E)′ is a soft -closed set contained in a soft open set (F, E)′. Hence cω(H, E)′⊂̃(F, E)′, i.e. (iω(H, E))′⊂̃(F, E)′. So (F, E)⊂̃iω(H, E). Conversely, Suppose that (F, E)⊂̃iω(H, E) for any soft closed set (F, E) whenever (F, E)⊂̃(H, E). Let (H, E)′⊂̃(G, E), where (G, E) is a soft open set. Then (G, E)′⊂̃(H, E) and (G, E)′ is soft closed. By assumption, (G, E)′⊂̃iω(H, E) and hence cω(H, E) = (iω(H, E))′⊂̃(G, E). Therefore, (H, E)′ is soft -closed and hence (H, E) is soft -open.

Theorem 2.13

Let ω be a soft weak structure on a soft topological space (X, τ, E). Then the following are equivalent:

1. For every soft open (G, E) of X, cω(G, E)⊂̃(G, E).

2. Every soft subset of X is soft -closed.

Proof

(1)⇒ (2) Let (F, E) be any soft subset of X where (F, E)⊂̃(G, E) and (G, E) is any soft open set. Then by (1), cω(G, E)⊂̃(G, E) and hence cω(F, E)⊂̃cω(G, E)⊂̃(G, E).

Thus (F, E) is soft -closed.

(2)⇒(1) Let (G, E) be any soft open set. Then by (2), (G, E) is soft -closed and hence cω(G, E)⊂̃(G, E).

Theorem 2.14

Let ω be a soft weak structure on a soft topological space (X, τ, E). If a soft set (H, E) is soft -open and if for a soft open set (G, E), iω(H, E)⊂̃(H, E)′⊂̃(G, E), then (G, E)′ = ∅̃.

Proof

Let (G, E) be a soft open set and iω(H, E)⊂̃(H, E)′⊂̃ (G, E) for a soft -open set (H, E). Then (G, E)′⊂̃(iω(H, E))′ ∩̃(H, E). This implies (G, E)′⊂̃cω(H, E) − (H, E). Since (H, E)′ is soft -closed, by Theorem 2.9, (G, E)′ = ∅̃.

Theorem 2.15

Let ω be a soft weak structure on a soft topological space (X, τ, E). If a soft set (H, E) is soft -open and iω(H, E)⊂̃(K, E)⊂̃(H, E), then (K, E) is soft -open.

Proof

We have (H, E)′⊂̃(K, E)′⊂̃cω(H, E)′. Since (H, E) is soft -closed, from Theorem 2.7, it follows that (K, E)′ is soft -closed, and hence (K, E) is soft -open.

3. Separation Axioms

In this section, we introduce the new separation axiom, namely soft ω-$T12$-space in soft topological space with a soft weak structure ω.

Definition 3.1

A soft topological space (X, τ, E) with a soft weak structure ω is called a soft ω-$T12$-space if for every soft -closed set (F, E) of X, cω(F, E) = (F, E)

Theorem 3.2

Let ω be a soft weak structure on X. A soft topological space (X, τ, E) is soft ω-T1 if (x, E) is soft ω-closed set for all XX.

Proof

Let x, yX such that Xy. Then (x, E)′ and (y, E)′ are soft ω-open sets such y ∈ (x, E)′, X/∈ (x, E)′ and y/∈ (y, E)′, X ∈ (y, E)′. Hence X is soft ω-T1.

Corollary 3.3

If the union of soft ω-open sets is soft ω-open set, then the converse of Theorem 3.2, is true.

Theorem 3.4

Let ω be a soft weak structure on a soft topological space (X, τ, E) with respect to E. Then the following statements are equivalent:

1. X is a soft ω-$T12$-space.

2. Every singleton is either soft closed or (x, E) = iω(x, E)

Proof

(1)⇒ (2). Suppose (x, E) is not a soft closed subset for some XX. Then (x, E)′ is not soft open, so if there is, then X is the only soft open set containing (x, E)′. Therefore (x, E)′ is soft -closed. Since X is soft ω-$T12$-space, cω(x, E)′ = (iω(x, E))′ = (x, E)′ and thus (x, E) = iω(x, E).

(2)⇒ (1) Let (F, E) be a soft -closed subset of X and Xcω(F, E). We show that X ∈ (F, E). If (x, E) is soft closed and X ∉ (F, E), then X ∈ (cω(F, E) − (F, E)). Then (x, E)⊂̃(F, E)′ and hence (F, E)⊂̃(x, E)′. Since (F, E) is a soft -closed set and (x, E)′ is a soft open, cω(F, E)⊂̃(x, E)′ and hence (x, E)⊂̃(cω(F, E))′. Therefore, (x, E)⊂̃cω(F, E) ∩̃(cω(F, E))′ = ∅̃. This is a contradiction. Therefore, X ∈ (F, E). If (x, E) = iω(x, E), since Xcω(F, E), then for every soft ω-open set (G, E) such that X ∈ (G, E), we have (G, E)∩̃(F, E) ≠ ∅̃. By assumption (x, E) = iω(x, E), we have (x, E) be an ω-open soft set and (x, E)∩̃(F, E) ≠ ∅̃. Hence X ∈ (F, E). Therefore, in both cases we have X ∈ (F, E). Therefore, cω(F, E) = (F, E) and hence X is a soft ω-$T12$-space.

Theorem 3.5

If the union of soft ω-open sets is soft ω-open set in a soft weak structure ω on a soft topological space, then every soft ω-T1-space is a soft ω-$T12$

-space.

Proof

Suppose that (X, τ, E) is a soft T1-space and the union of soft ω-open sets is soft ω-open set. It suffices to show that a set which is not soft ω-closed also is not soft -closed set. Let (F, E) is not soft ω-closed. Let Xcω(F, E) − (F, E). Then (x, E)⊂̃cω(F, E)−(F, E) and (x, E) is a nonempty soft ω-closed set in X by Corollary 3.3. Hence, by Theorem 2.9, (F, E) is not soft -closed.

The next example shows that the converse of the above theorem is not true in general.

Example 3.6

Let X = {h1, h2, h3}, E = {e1, e2} and τ = {∅̃, X̃, (F1, E), (F2, E)} where

$F1(e1)={h1},F1(e2)={h1};F2(e1)={h1,h2},F2(e2)=X;$

Let ω = {∅̃, (F1, E), (F2, E)} be an ω over X with respect to E. Then (X, τ, E) is soft ω-$T12$-space but not soft ω-T1.

Theorem 3.7

Let ω be a soft weak structure on a soft topological space (X, τ, E). If X is soft ω-T0, then for each x, yX such that Xy, we have cω(x, E) ≠̃cω(y, E).

Proof

Let X be a soft ω-T0 and x, yX such that x≠̃y. Then there exists soft ω-open set (F, E) such that X ∈ (F, E) and y/∈ (F, E). Therefore (F, E)′ is soft ω-closed set such that X/∈ (F, E)′ and y ∈ (F, E)′. Since cω(y, E) is the intersection of all soft ω-closed subsets that contain y, then cω(y, E)⊂̃(F, E)′ and hence X/cω(y, E). Thus cω(x, E) ≠̃cω(y, E)

Corollary 3.8

If cω(x, E) is a soft ω-closed set for each XX, and if for each distinct x, yX, cω(x, E) ≠̃cω(y, E), then X is soft ω-T0.

Proof

For each distinct x, yX, since cω(x, E) ≠̃cω(y, E), there exists some zX such that zcω(x, E) and z/cω(y, E). If Xcω(y, E), then cω(x, E)⊂̃cω(y, E) which is a contradiction since z/cω(y, E). Thus (cω(y, E))′ is soft ω -open set such that X ∈ (cω(y, E))′ and y/∈ (cω(y, E))′. Hence X is soft ω-T0.

Conflict of Interest

No potential conflict of interest relevant to this article was reported.

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Biographies

Hanan S. Al-Saadi is currently an associate Professor in Umm Al-Qura University. She received the Ph.D. degree in Pure Mathematics. Her primary research areas are Fuzzy topology and General topology.

Phone: +966-012-5472104

E-mail: hasa112@hotmail.com

Won Keun Min is currently a professor in Kangwon National University. Chunchon, Korea. His main research interests include fuzzy topology and general topology.

Phone: +82-33-250-8419

Fax: +82-33-252-7298

E-mail: wkmin@cc.kangwon.ac.kr

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