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## Original Article

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International Journal of Fuzzy Logic and Intelligent Systems 2020; 20(2): 119-123

Published online June 25, 2020

https://doi.org/10.5391/IJFIS.2020.20.2.119

© The Korean Institute of Intelligent Systems

## On Soft ω-Structures Defined by Soft Sets

Won Keun Min

Department of Mathematics, Kangwon National University, Chuncheon, Korea

Correspondence to :
Won Keun Min (wkmin@kangwon.ac.kr)

Received: October 15, 2018; Revised: April 10, 2020; Accepted: April 23, 2020

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 noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this work, we introduce the notion of soft w-structure and investigate some basic properties of this new structure by using the concept of soft set. Moreover, we study the notions of soft wT0 (soft wT1, soft wT2).

Keywords: Soft set, Soft topology, Soft ω-structure, Soft ω-T0 (ω-T1, ω-T2).

### 1. Introduction and Preliminaries

In 1999, Molodtsov [1] initiated the notion of soft set theory as a new mathematical tool which is free from the complex problems. 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 the author [5] corrected some their results. Zorlutuna et al. [6] continued to study the properties of soft topological spaces by defining the concepts of interior and soft neighborhoods in soft topological spaces. In 2011, Cagman et al. [7] defined soft topological spaces by modifying the soft set. Also, Roy and Samanta [8] strengthen the definition of the soft topological spaces presented in [7].

In 2017, with the aim of generalizing the notion of soft topology, Zakari et al. [9] introduced a soft weak structure. Recently, Al-Saadi and Min [10] investigated the notion of soft generalized closed sets in a soft weak structure.

Meanwhile, Min and Kim [11] introduced a new notion called weak structures as the following: Let X be a non-empty set and P(X) be its power set. A structure ω on X is called a weak structure on X if and only if (i) φω: (ii) For U, Vω, UVω.

In this work, by applying the notion of weak structure in [11], we want to introduce the new notion of soft w-structure which is a generalized soft topological structure defined by Shabir and Naz [10] and a stronger structure than soft weak structure defined by Zakari et al. [9]. And we investigate some basic properties of this new class by using the notion of soft set. Moreover, we study the notions of soft wT0 (soft wT1, soft wT2) and some properties of such notions. From now on, let X be a non-empty common universe, E a set of parameters, and P(X) denote the power set of X.

### 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 ( [2])

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

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

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

### Definition 1.3 ( [2])

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

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

• (F, A)=(G, B) iff (F, A) ⊂̃ (G, B) and (G, B) ⊂̃ (F, A).

• (F, A)∪̃(G, B)=(H, C) where C = AB and

H(e)={F(e),if eA-B,G(e),if eB-A,F(e)G(e),if eAB,

for all eC.

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

• x ∈ (F, A) where xX iff xF(e) for all eA and x ∉ (F, A) whenever xF(e) for some eA.

### Definition 1.4 ([12])

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:

• ̃, belong to τ.

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

• 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 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.

### Definition 2.1

Let sw be the collection of soft sets over X. Then sw is called a soft w-structure on X if sw satisfies the following axioms:

• ̃, belong to sw.

• The intersection of any two soft sets in sw belongs to sw.

The triple (X, sw, E) is called a soft w-space over X. The member of sw is said to be soft w-open in X. A soft set (F, E) over X is said to be soft w-closed in X if its relative complement (F, E)′ belongs to sw.

### Remark 2.2

Let sw be a soft w-structure over X. The soft w-structure sw is a kind of generalized soft topology and a stronger structure than a soft weak structure defined by Zakari et al. [9] as the following: Let X be a non-empty set and E a set of parameters. A collection ω of soft sets defined over X with respect to E is called a soft weak structure [9] iff ̃ ∈ ω.

### Example 2.3

Let X = {h1, h2, h3, h4}, E = {e1, e2} and sw = {̃, , (F1, E), (F2, E), (F3, E)}, where

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

Then sw is a soft w-structure over X with respect to E but not a soft topology.

### Definition 2.4

Let sw be a soft w-structure over X with respect to E. For a soft set (F, E) over X, the soft w-closure of (F, E) (simply, csw(F, E)) and the soft w-interior of (F, E) (simply, csw(F, E)) are defined as the following:

• isw(F, E)= ∪̃{(G, E) : (G, E) ⊂̃ (F, E), (G, E) ∈ sw}.

• csw(F, E)= ∩̃{(H, E) : (F, E) ⊂̃ (H, E), (H, E)′ ∈ sw}.

### Theorem 2.5

Let sw be a soft w-structure over X with respect to the parameters set E and (F, E) a soft set. If there exists a soft w-open set (G, E) such that x ∈ (G, E) ⊂̃ (F, E), then xisw(F, E)

Proof

It is obvious.

### Example 2.6

As in Example 2.3, consider the soft w-structure sw over X with respect to E and a soft set (F4, E) as follows:

F4(e1)={h1,h2,h3},         F4(e2)={h1,h2,h3}.

Then (F4, E) = isw(F4, E). For h3isw(F4, E), there is no a soft w-open set containing h3 in sw. So the converse of Theorem 2.5 is not always true.

### Theorem 2.7

Let sw be a soft w-structure over X with respect to the parameters set E and (F, E) a soft set. If xcsw(F, E), then (G, E)∩̃(F, E) ≠ ̃ for all (G, E) ∈ sw such that x ∈ (G, E).

Proof

Let xcsw(F, E). Suppose that there exists an element (G, E) ∈ sw such that x ∈ (G, E) and (F, E)∩̃(G, E) = ̃. Then (F, E) ⊂ (G, E)′, so csw(F, E) ⊂̃ (G, E)′ and xcsw(F, E). So it is a contradiction.

### Example 2.8

Let X = {h1, h2, h3}, E = {e1, e2} and sw = {̃, , (F1, E), (F2, E), (F3, E)} where

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

Then sw is a soft w-structure over X with respect to E. Consider a soft set (F4, E) defined as:

F4(e1)={h1},         F4(e2)={h3}.

Since (F4, E) is soft w-closed, (F4, E) = csw(F4, E). For h1X, (F2, E) is the only soft w-open set and (F4, E)∩̃(F2, E) ≠ ̃, however, h1 ∉̃ csw(F4, E). So the converse of Theorem 2.7 is not always true.

### Theorem 2.9

Let sw be a soft w-structure defined over X with respect to the parameters set E and (F, E) be a soft set.

• If (F, E) is a soft w-open set, then (F, E) = isw(F, E).

• If (F, E) is a soft w-closed set, then (F, E) = csw(F, E).

Proof

From the definitions of soft w-interior and soft w-closure, it is obvious.

But the converses in Theorem 2.9 are not always true as shown the next example.

### Example 2.10

Let X = {h1, h2, h3}, E = {e1, e2} and sw = {̃, , (F1, E), (F2, E), (F3, E), (F4, E), (F5, E)}, where

F1(e1)={h3},F1(e2)={h2};F2(e1)={h2,h3},F2(e2)={h1,h2};F3(e1)={h1,h2},F3(e2)={h1,h3};F4(e1)={h1},F4(e2)={h3};F5(e1)={h2},F4(e2)={h1};F6(e1)={h1,h3},F4(e2)={h2,h3};

Then sw is a soft w-structure over X with respect to E. For a soft set (F5, E), csw(F5, E) = (F5, E) but (F5, E) is not soft w-closed. And, for a soft set (F6, E), isw(F6, E) = (F6, E) but (F6, E) is not soft w-open.

### Theorem 2.11

Let sw be a soft w-structure over X with respect to E. Let (F, E) and (G, E) be two soft sets over X. Then:

• isw(F, E) ⊂̃ (F, E).

• If (F, E) ⊂̃ (G, E), then isw(F, E) ⊂̃ isw(G, E).

• isw((F, E)∩̃(G, E)) = isw(F, E)∩̃isw(G, E).

• isw(isw(F, E)) = isw(F, E).

Proof

(1) and (2) are obvious.

(3) It is obvious that isw((F, E)∩̃(G, E)) ⊂̃ isw(F, E)∩̃isw (G, E) from (2). For soft w-open sets (U, E) ⊂̃ (F, E) and (V, E) ⊂̃ (G, E), (U, E)∩̃(V, E) is a soft w-open set contained in (F, E)∩̃(G, E). This implies that

isw(F,E)˜isw(G,E)˜isw((F,E))˜(G,E)).

(4) From (1), it follows isw(isw(F, E))⊆̃isw(F, E). For any soft w-open set (U, E) such that (U, E)⊆̃isw(F, E), (U, E) = isw(U, E)⊆̃isw(isw(F, E)), and so isw(F, E)⊆̃isw(isw(F, E)). Consequently, we have isw(isw(F, E)) = isw(F, E).

### Theorem 2.12

Let sw be a soft w-structure defined over X with respect to E. If (F, E) and (G, E) are two soft sets over X, then:

• (F, E) ⊂̃ csw(F, E).

• If (F, E) ⊂̃ (G, E), then csw(F, E) ⊂̃ csw(G, E).

• csw(F, E)∪̃csw(G, E) = csw((F, E)∪̃(G, E)).

• csw(csw(F, E)) = csw(G, E).

• isw(F, E)′=(csw(F, E))′ and csw(F, E)′=(isw(F, E))′.

Proof

It is similar to the proof of Theorem 2.11.

Now, we introduce the separation axioms in soft w-space with a soft w-structure sw.

### Definition 2.13

Let sw be a soft w-structure over X with respect to E. A soft w-space (X, sw, E) is called:

• wT0 if for each x, yX such that xy, there exists a soft w-open set (F, E) such that x ∈ (F, E) and y ∉ (F, E) or x ∉ (F, E) and y ∈ (F, E).

• wT1 if for each x, yX such that xy, there exist soft w-open sets (F, E) and (G, E) such that x ∈ (F, E) and y ∉ (F, E) and x ∉ (G, E) and y ∈ (G, E).

• wT2 if for each x, yX such that xy, there exist soft w-open sets (F, E) and (G, E) such that x ∈ (F, E), y ∈ (G, E) and (F, E)∩̃(G, E) = ̃.

We have the following diagram:

soft w-T2soft w-T1soft w-T0.

### Example 2.14

Let X = {h1, h2, h3}, E = {e1, e2} and sw = {̃, , (F1, E), (F2, E), (F3, E), (F4, E), (F5, E), (F6, E)}, where

F1(e1)={h3},F1(e2)={h3};F2(e1)={h1,h2},F3(e2)={h1,h3};F3(e1)={h2,h3},F2(e2)={h1,h3};F4(e1)={h2},F4(e2)={h1};F5(e1)={h3},F4(e2)=;F6(e1)=,F4(e2)={h3}.

Then sw is a soft w-structure over X with respect to E. It is obviously a soft wT1 space. For h1, h2X, (F2, E) and (F3, E) are unique soft w-open sets of h1, h2, respectively. But (F2, E)∩̃(F3, E) ≠ ̃. So (X, sw, E) is not soft wT2.

### Example 2.15

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

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

Then sw is a soft w-structure over X with respect to E. It is obviously a soft wT0 space but it is not soft wT1.

Let sw be a soft w-structure over X with respect to E. A soft w-space (X, sw, E) is called relative soft wT0 if for each x, yX such that xy, there exists a soft w-open set (F, E) such that x ∈ (F, E) and y ∈ (F, E)′ or x ∈ (F, E)′ and y ∈ (F, E).

### Theorem 2.16

Let sw be a soft w-structure on X. If X is a relative soft wT0 space, then for each x, yX such that xy, we have csw(x, E) ≠ csw(y, E).

Proof

Let X be a relative soft wT0 and x, yX such that xy. Then there exists a soft w-open set (F, E) such that x ∈ (F, E) and y ∈ (F, E)′. Therefore (F, E)′ is a soft w-closed set such that x ∉ (F, E)′ and y ∈ (F, E)′. Since csw(y, E) is the intersection of all soft w-closed subsets containing (y, E), csw(y, E) ⊂̃ (F, E)′ and hence xcsw(y, E). Thus csw(x, E) ≠ csw(y, E).

### Theorem 2.17

Let sw be a soft w-structure on X. If ycsw(x, E), then for each soft w-open set (G, E) containing y, there exists a parameter eE such that xG(e).

Proof

Let y ∈ (csw(x, E)). Then by Theorem 2.7, (G, E)∩̃(x, E) ≠ ̃ for all (G, E) ∈ sw such that y ∈ (G, E). Since (G, E)∩̃(x, E) ≠ ̃, there exists a parameter eE such that xG(e).

### Theorem 2.18

Let sw be a soft weak structure on X. A soft w-space (X, sw, E) is soft wT1 if (x, E) is soft w-closed set for all xX.

Proof

Let x, yX such that xy. Then (x, E)′ and (y, E)′ are soft w-open sets such y ∈ (x, E)′, x ∈ (x, E)′ and y ∉ (y, E)′, x ∈ (y, E)′. Hence X is soft wT1.

### 3. Conclusions

The author introduced the notion of soft w-structure and investigated some basic properties of this new structure. In the next research, the author will introduce the associated soft w-structures induced by soft topologies and study the relationship between soft w-structures and associated soft w-structure induced by soft topologies.

### References

1. D. Molodtsov, "Soft set theory—First results," Computers & Mathematics with Applications, vol. 37, no. 4–5, pp. 19-31, 1999. https://doi.org/10.1016/S0898-1221(99)00056-5
2. P. K. Maji, R. Biswas, A. R. Roy, "Soft set theory," Computers & Mathematics with Applications, vol. 45, no. 4–5, pp. 555-562, 2003. https://doi.org/10.1016/S0898-1221(03)00016-6
3. D. Pie, D. Miao, "From soft sets to information systems," in Proceedings of 2005 IEEE International Conference on Granular Computing, Beijing, China, , pp. 617-621. https://doi.org/10.1109/GRC.2005.1547365
4. M. Shabir, M. Naz, "On soft topological spaces," Computers & Mathematics with Applications, vol. 61, no. 7, pp. 1786-1799, 2011. https://doi.org/10.1016/j.camwa.2011.02.006
5. W. K. Min, "A note on soft topological spaces," Computers & Mathematics with Applications, vol. 62, no. 9, pp. 3524-3528, 2011. https://doi.org/10.1016/j.camwa.2011.08.068
6. I. Zorlutuna, M. Akdag, W. K. Min, S. Atmaca, "Remarks on soft topological spaces," Annals of Fuzzy Mathematics and Informatics, vol. 3, no. 2, pp. 171-185, 2012.
7. N. Cagman, S. Karatas, S. Enginoglu, "Soft topology," Computers & Mathematics with Applications, vol. 62, no. 1, pp. 351-358, 2011. https://doi.org/10.1016/j.camwa.2011.05.016
8. S. Roy, T. K. Samanta, "An introduction of a soft topological spaces," in Proceedings of UGC Sponsored National Seminar on Recent Trends in Fuzzy Set Theory, Rough Set Theory and Soft Set Theory, Howrah, India, , pp. 9-12.
9. A. H. Zakari, A. Ghareeb, S. Omran, "On soft weak structures," Soft Computing, vol. 21, no. 10, pp. 2553-2559, 2017. https://doi.org/10.1007/s00500-016-2136-8
10. H. S. Al-Saadi, W. K. Min, "On soft generalized closed sets in a soft topological space with a soft weak structure," International Journal of Fuzzy Logic and Intelligent Systems, vol. 17, no. 4, pp. 323-328, 2017. https://doi.org/10.5391/IJFIS.2017.17.4.323
11. Y. K. Kim, W. K. Min, "On weak structures and w-spaces," Far East Journal of Mathematical Sciences, vol. 97, no. 5, pp. 549-561, 2015. http://dx.doi.org/10.17654/FJMSJul2015_549_561
12. M. I. Ali, F. Feng, X. Liu, W. K. Min, M. Shabir, "On some new operations in soft set theory," Computers & Mathematics with Applications, vol. 57, no. 9, pp. 1547-1553, 2009. https://doi.org/10.1016/j.camwa.2008.11.009

### Biography

Won Keun Min received the M.S. and the Ph.D. degrees in mathematics from Korea University, Seoul, Korea in 1983 and 1987, respectively. He is currently a professor in the Department of Mathematics, Kangwon National University. His research interests include general topology, fuzzy topology and soft set theory.

E-mail: wkmin@kangwon.ac.kr

### Article

#### Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2020; 20(2): 119-123

Published online June 25, 2020 https://doi.org/10.5391/IJFIS.2020.20.2.119

## On Soft ω-Structures Defined by Soft Sets

Won Keun Min

Department of Mathematics, Kangwon National University, Chuncheon, Korea

Correspondence to:Won Keun Min (wkmin@kangwon.ac.kr)

Received: October 15, 2018; Revised: April 10, 2020; Accepted: April 23, 2020

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 noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

In this work, we introduce the notion of soft w-structure and investigate some basic properties of this new structure by using the concept of soft set. Moreover, we study the notions of soft wT0 (soft wT1, soft wT2).

Keywords: Soft set, Soft topology, Soft ω-structure, Soft ω-T0 (ω-T1, ω-T2).

### 1. Introduction and Preliminaries

In 1999, Molodtsov [1] initiated the notion of soft set theory as a new mathematical tool which is free from the complex problems. 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 the author [5] corrected some their results. Zorlutuna et al. [6] continued to study the properties of soft topological spaces by defining the concepts of interior and soft neighborhoods in soft topological spaces. In 2011, Cagman et al. [7] defined soft topological spaces by modifying the soft set. Also, Roy and Samanta [8] strengthen the definition of the soft topological spaces presented in [7].

In 2017, with the aim of generalizing the notion of soft topology, Zakari et al. [9] introduced a soft weak structure. Recently, Al-Saadi and Min [10] investigated the notion of soft generalized closed sets in a soft weak structure.

Meanwhile, Min and Kim [11] introduced a new notion called weak structures as the following: Let X be a non-empty set and P(X) be its power set. A structure ω on X is called a weak structure on X if and only if (i) φω: (ii) For U, Vω, UVω.

In this work, by applying the notion of weak structure in [11], we want to introduce the new notion of soft w-structure which is a generalized soft topological structure defined by Shabir and Naz [10] and a stronger structure than soft weak structure defined by Zakari et al. [9]. And we investigate some basic properties of this new class by using the notion of soft set. Moreover, we study the notions of soft wT0 (soft wT1, soft wT2) and some properties of such notions. From now on, let X be a non-empty common universe, E a set of parameters, and P(X) denote the power set of X.

### 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 ( [2])

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

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

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

### Definition 1.3 ( [2])

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

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

• (F, A)=(G, B) iff (F, A) ⊂̃ (G, B) and (G, B) ⊂̃ (F, A).

• (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.

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

• x ∈ (F, A) where xX iff xF(e) for all eA and x ∉ (F, A) whenever xF(e) for some eA.

### Definition 1.4 ([12])

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:

• ̃, belong to τ.

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

• 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 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.

### Definition 2.1

Let sw be the collection of soft sets over X. Then sw is called a soft w-structure on X if sw satisfies the following axioms:

• ̃, belong to sw.

• The intersection of any two soft sets in sw belongs to sw.

The triple (X, sw, E) is called a soft w-space over X. The member of sw is said to be soft w-open in X. A soft set (F, E) over X is said to be soft w-closed in X if its relative complement (F, E)′ belongs to sw.

### Remark 2.2

Let sw be a soft w-structure over X. The soft w-structure sw is a kind of generalized soft topology and a stronger structure than a soft weak structure defined by Zakari et al. [9] as the following: Let X be a non-empty set and E a set of parameters. A collection ω of soft sets defined over X with respect to E is called a soft weak structure [9] iff ̃ ∈ ω.

### Example 2.3

Let X = {h1, h2, h3, h4}, E = {e1, e2} and sw = {̃, , (F1, E), (F2, E), (F3, E)}, where

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

Then sw is a soft w-structure over X with respect to E but not a soft topology.

### Definition 2.4

Let sw be a soft w-structure over X with respect to E. For a soft set (F, E) over X, the soft w-closure of (F, E) (simply, csw(F, E)) and the soft w-interior of (F, E) (simply, csw(F, E)) are defined as the following:

• isw(F, E)= ∪̃{(G, E) : (G, E) ⊂̃ (F, E), (G, E) ∈ sw}.

• csw(F, E)= ∩̃{(H, E) : (F, E) ⊂̃ (H, E), (H, E)′ ∈ sw}.

### Theorem 2.5

Let sw be a soft w-structure over X with respect to the parameters set E and (F, E) a soft set. If there exists a soft w-open set (G, E) such that x ∈ (G, E) ⊂̃ (F, E), then xisw(F, E)

Proof

It is obvious.

### Example 2.6

As in Example 2.3, consider the soft w-structure sw over X with respect to E and a soft set (F4, E) as follows:

$F4(e1)={h1,h2,h3}, F4(e2)={h1,h2,h3}.$

Then (F4, E) = isw(F4, E). For h3isw(F4, E), there is no a soft w-open set containing h3 in sw. So the converse of Theorem 2.5 is not always true.

### Theorem 2.7

Let sw be a soft w-structure over X with respect to the parameters set E and (F, E) a soft set. If xcsw(F, E), then (G, E)∩̃(F, E) ≠ ̃ for all (G, E) ∈ sw such that x ∈ (G, E).

Proof

Let xcsw(F, E). Suppose that there exists an element (G, E) ∈ sw such that x ∈ (G, E) and (F, E)∩̃(G, E) = ̃. Then (F, E) ⊂ (G, E)′, so csw(F, E) ⊂̃ (G, E)′ and xcsw(F, E). So it is a contradiction.

### Example 2.8

Let X = {h1, h2, h3}, E = {e1, e2} and sw = {̃, , (F1, E), (F2, E), (F3, E)} where

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

Then sw is a soft w-structure over X with respect to E. Consider a soft set (F4, E) defined as:

$F4(e1)={h1}, F4(e2)={h3}.$

Since (F4, E) is soft w-closed, (F4, E) = csw(F4, E). For h1X, (F2, E) is the only soft w-open set and (F4, E)∩̃(F2, E) ≠ ̃, however, h1 ∉̃ csw(F4, E). So the converse of Theorem 2.7 is not always true.

### Theorem 2.9

Let sw be a soft w-structure defined over X with respect to the parameters set E and (F, E) be a soft set.

• If (F, E) is a soft w-open set, then (F, E) = isw(F, E).

• If (F, E) is a soft w-closed set, then (F, E) = csw(F, E).

Proof

From the definitions of soft w-interior and soft w-closure, it is obvious.

But the converses in Theorem 2.9 are not always true as shown the next example.

### Example 2.10

Let X = {h1, h2, h3}, E = {e1, e2} and sw = {̃, , (F1, E), (F2, E), (F3, E), (F4, E), (F5, E)}, where

$F1(e1)={h3},F1(e2)={h2};F2(e1)={h2,h3},F2(e2)={h1,h2};F3(e1)={h1,h2},F3(e2)={h1,h3};F4(e1)={h1},F4(e2)={h3};F5(e1)={h2},F4(e2)={h1};F6(e1)={h1,h3},F4(e2)={h2,h3};$

Then sw is a soft w-structure over X with respect to E. For a soft set (F5, E), csw(F5, E) = (F5, E) but (F5, E) is not soft w-closed. And, for a soft set (F6, E), isw(F6, E) = (F6, E) but (F6, E) is not soft w-open.

### Theorem 2.11

Let sw be a soft w-structure over X with respect to E. Let (F, E) and (G, E) be two soft sets over X. Then:

• isw(F, E) ⊂̃ (F, E).

• If (F, E) ⊂̃ (G, E), then isw(F, E) ⊂̃ isw(G, E).

• isw((F, E)∩̃(G, E)) = isw(F, E)∩̃isw(G, E).

• isw(isw(F, E)) = isw(F, E).

Proof

(1) and (2) are obvious.

(3) It is obvious that isw((F, E)∩̃(G, E)) ⊂̃ isw(F, E)∩̃isw (G, E) from (2). For soft w-open sets (U, E) ⊂̃ (F, E) and (V, E) ⊂̃ (G, E), (U, E)∩̃(V, E) is a soft w-open set contained in (F, E)∩̃(G, E). This implies that

$isw(F,E)∩˜isw(G,E)⊂˜isw((F,E))∩˜(G,E)).$

(4) From (1), it follows isw(isw(F, E))⊆̃isw(F, E). For any soft w-open set (U, E) such that (U, E)⊆̃isw(F, E), (U, E) = isw(U, E)⊆̃isw(isw(F, E)), and so isw(F, E)⊆̃isw(isw(F, E)). Consequently, we have isw(isw(F, E)) = isw(F, E).

### Theorem 2.12

Let sw be a soft w-structure defined over X with respect to E. If (F, E) and (G, E) are two soft sets over X, then:

• (F, E) ⊂̃ csw(F, E).

• If (F, E) ⊂̃ (G, E), then csw(F, E) ⊂̃ csw(G, E).

• csw(F, E)∪̃csw(G, E) = csw((F, E)∪̃(G, E)).

• csw(csw(F, E)) = csw(G, E).

• isw(F, E)′=(csw(F, E))′ and csw(F, E)′=(isw(F, E))′.

Proof

It is similar to the proof of Theorem 2.11.

Now, we introduce the separation axioms in soft w-space with a soft w-structure sw.

### Definition 2.13

Let sw be a soft w-structure over X with respect to E. A soft w-space (X, sw, E) is called:

• wT0 if for each x, yX such that xy, there exists a soft w-open set (F, E) such that x ∈ (F, E) and y ∉ (F, E) or x ∉ (F, E) and y ∈ (F, E).

• wT1 if for each x, yX such that xy, there exist soft w-open sets (F, E) and (G, E) such that x ∈ (F, E) and y ∉ (F, E) and x ∉ (G, E) and y ∈ (G, E).

• wT2 if for each x, yX such that xy, there exist soft w-open sets (F, E) and (G, E) such that x ∈ (F, E), y ∈ (G, E) and (F, E)∩̃(G, E) = ̃.

We have the following diagram:

$soft w-T2⇒soft w-T1⇒soft w-T0.$

### Example 2.14

Let X = {h1, h2, h3}, E = {e1, e2} and sw = {̃, , (F1, E), (F2, E), (F3, E), (F4, E), (F5, E), (F6, E)}, where

$F1(e1)={h3},F1(e2)={h3};F2(e1)={h1,h2},F3(e2)={h1,h3};F3(e1)={h2,h3},F2(e2)={h1,h3};F4(e1)={h2},F4(e2)={h1};F5(e1)={h3},F4(e2)=∅;F6(e1)=∅,F4(e2)={h3}.$

Then sw is a soft w-structure over X with respect to E. It is obviously a soft wT1 space. For h1, h2X, (F2, E) and (F3, E) are unique soft w-open sets of h1, h2, respectively. But (F2, E)∩̃(F3, E) ≠ ̃. So (X, sw, E) is not soft wT2.

### Example 2.15

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

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

Then sw is a soft w-structure over X with respect to E. It is obviously a soft wT0 space but it is not soft wT1.

Let sw be a soft w-structure over X with respect to E. A soft w-space (X, sw, E) is called relative soft wT0 if for each x, yX such that xy, there exists a soft w-open set (F, E) such that x ∈ (F, E) and y ∈ (F, E)′ or x ∈ (F, E)′ and y ∈ (F, E).

### Theorem 2.16

Let sw be a soft w-structure on X. If X is a relative soft wT0 space, then for each x, yX such that xy, we have csw(x, E) ≠ csw(y, E).

Proof

Let X be a relative soft wT0 and x, yX such that xy. Then there exists a soft w-open set (F, E) such that x ∈ (F, E) and y ∈ (F, E)′. Therefore (F, E)′ is a soft w-closed set such that x ∉ (F, E)′ and y ∈ (F, E)′. Since csw(y, E) is the intersection of all soft w-closed subsets containing (y, E), csw(y, E) ⊂̃ (F, E)′ and hence xcsw(y, E). Thus csw(x, E) ≠ csw(y, E).

### Theorem 2.17

Let sw be a soft w-structure on X. If ycsw(x, E), then for each soft w-open set (G, E) containing y, there exists a parameter eE such that xG(e).

Proof

Let y ∈ (csw(x, E)). Then by Theorem 2.7, (G, E)∩̃(x, E) ≠ ̃ for all (G, E) ∈ sw such that y ∈ (G, E). Since (G, E)∩̃(x, E) ≠ ̃, there exists a parameter eE such that xG(e).

### Theorem 2.18

Let sw be a soft weak structure on X. A soft w-space (X, sw, E) is soft wT1 if (x, E) is soft w-closed set for all xX.

Proof

Let x, yX such that xy. Then (x, E)′ and (y, E)′ are soft w-open sets such y ∈ (x, E)′, x ∈ (x, E)′ and y ∉ (y, E)′, x ∈ (y, E)′. Hence X is soft wT1.

### 3. Conclusions

The author introduced the notion of soft w-structure and investigated some basic properties of this new structure. In the next research, the author will introduce the associated soft w-structures induced by soft topologies and study the relationship between soft w-structures and associated soft w-structure induced by soft topologies.

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