Strong Form of Soft Semi-Open Sets in Soft Topological Spaces

Samer Al Ghour

Department of Mathematics and Statistics, Jordan University of Science and Technology, Irbid, Jordan

Received April 19, 2021; Revised June 3, 2021; Accepted June 8, 2021.

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
- Soft
*ω*-open sets as a class of soft sets that lies strictly between soft open sets and soft semi-open sets is introduced. The natural properties of soft_{s}*ω*-open sets are described. Using soft_{s}*ω*-open sets, soft_{s}*ω*-closure and soft_{s}*ω*-interior as new soft operators are defined and investigated. Furthermore, the relationships regarding generated soft topological spaces and generated topological spaces are studied._{s} **Keywords**: Soft*ω*-open sets, Soft semi-open sets, Soft*ω*-open, Generated soft topology_{s}

- 1. Introduction
Throughout this paper, we follow the notions and terminologies as used in [1] and [2], and for simplicity, STS stands for soft topological space. For the purpose of dealing with uncertain objects, Molodtsov [3] introduced soft sets in 1999. Let

*Y*be a universal set and*B*be a set of parameters. A soft set over*Y*relative to*B*is a function*H*:*B*→*℘*(*Y*). The family of all soft sets over*Y*relative to*B*is denoted by*SS*(*Y, B*). Let*H*∈*SS*(*Y, B*), then*H*is a countable soft set if*H*(*b*) is a countable set for all*b*∈*B*. The family of all members of*SS*(*Y, B*) that are countable is denoted by*CSS*(*Y, B*). Furthermore, the null soft set and the absolute soft set are denoted by 0and 1_{B}, respectively. The authors in [4] define the notion of STSs as follows: The triplet (_{B}*Y, τ, B*), where*τ*⊆*SS*(*Y, B*) is a STS if 0and 1_{B}∈_{B}*τ, τ*is closed under finite soft intersection, and*τ*is closed under arbitrary soft union. If (*Y, τ, B*) is a STS, then the members of*τ*are called soft open sets and their soft complements are called soft closed sets. Topologists recently have applied various topological concepts to soft topological spaces ([1,2,5–25]).Chen [26] introduced the concept of soft semi-open sets, a weaker form of soft open sets in STSs. This paper was followed by many more papers about soft semi-open sets and their modifications. In this paper, we introduce soft

*ω*-open sets as a class of soft sets that lies strictly between soft open sets and soft semi-open sets. We present the natural properties of soft_{s}*ω*-open sets. Furthermore, using_{s}*ω*-open sets, we define and investigate soft_{s}*ω*-closure and_{s}*ω*-interior as new soft topological operators. We also compare our proposed concepts and their corresponding previous analogous concepts in Theorems 2.2, 2.17, 3.3, and 3.19, with the comparisons supported by examples. We present several sufficient conditions for the equivalence between the proposed concepts and their corresponding previous analogous concepts. We next discuss relationships between relative topological spaces and generated soft topological spaces. Finally, we raise an open question related to soft_{s}*ω*-open._{s}In [27,28], the authors showed that soft sets are a class of special information systems. This insight constitutes our motivation to study the structures of soft sets for information systems. Therefore, this paper does not only form the theoretical basis for further applications of soft topology, such as soft

*ω*mappings and soft_{s-}*ω*-compactness, but also comments on the future development of information systems._{s}The following definitions and results will be used in the sequel:

### Definition 1.1

Let (

*X*, ℑ) be a topological space and let*D*⊆*X*. The*D*is called(a) [29] semi-open if there is

*V*∈ ℑ such that*V*⊆*D*⊆*Cl*_{ℑ}(*V*).*SO*(*X*, ℑ) will denote the family of all semi-open sets in (*X*, ℑ).(b) [30]

*ω*-open if there is_{s}*V*∈ ℑ such that*V*⊆*D*⊆*Cl*_{ℑω}(*V*).*ω*(_{s}*X*, ℑ) will denote the family of all*ω*-open sets in (_{s}*X*, ℑ).### Definition 1.2 [26]

Let (

*X, τ, A*) be a STS and let*K*∈*SS*(*X, A*). Then(a)

*K*is called a soft semi-open set in (*X, τ, A*) if there exists*F*∈*τ*such that*F*⊆̃*K*⊆̃*Cl*(_{τ}*F*).*SO*(*X, τ, A*) will denote the family of all soft semi-open sets in (*X, τ, A*).(b)

*K*is called a soft semi-closed set in (*X, τ, A*) if 1–_{A}*K*∈*SO*(*X, τ, A*).### Definition 1.3 [26]

Let (

*X, τ, A*) be a STS and*H*∈*SS*(*X*,*A*).(a) The soft semi-closure of

*H*in (*X, τ, A*) is denoted by*S*-*Cl*(_{τ}*H*) and defined by*S*-*Cl*(_{τ}*H*) = ∩̃ {*M*:*M*is soft semi-closed in (*X, τ, A*) and*H*⊆̃*M*}.(b) The soft semi-interior of

*H*in (*X, τ, A*) is denoted by*S*-*int*(_{τ}*H*) and defined by*S*-*int*(_{τ}*H*) = ∪̃ {*K*:*K*is soft semi-open in (*X, τ, A*) and*K*⊆̃*H*}.As defined in [2], a STS (

*X, τ, A*) is called soft anti-locally countable if for every*F*∈*τ*– {0},_{A}*F*∉*CSS*(*X, A*).### Proposition 1.4 [2]

Let (

*X, τ, A*) be soft anti-locally countable. Then for all*G*∈*τ*,_{ω}*Cl*(_{τ}*G*) =*Cl*_{τ}_{ω}(*G*). As defined in [2], a STS (*X, τ, A*) is called soft locally countable if for*a*∈_{x}*SP*(*X, A*) there exists*G*∈*CSS*(*X, A*) ∩*τ*such that*a*∊̃_{x}*G*.### Proposition 1.5 [2]

A STS (

*X, τ, A*) is soft locally countable if and only if (*X, τ*_{ω}*, A*) is a discrete STS.### Proposition 1.6 [2]

For any STS (

*X, τ, A*) we have*τ*= (_{ω}*τ*)_{ω}._{ω}### Proposition 1.7 [26]

Let (

*X, τ, A*) be a STS and*H*∈*SS*(*X, A*). Then(a)

*S*-*Cl*(_{τ}*H*) is the smallest soft semi-closed in (*X, τ, A*) containing*H*.(b)

*H*is soft semi-closed in (*X, τ, A*) if and only if*H*=*S-Cl*(_{τ}*H*).

- 2. Soft
ω -Open sets_{s} ### Definition 2.1

Let (

*X, τ, A*) be a STS and let*K*∈*SS*(*X, A*). Then*K*is called a soft*ω*-open set in (_{s}*X, τ, A*) if there exists*F*∈*τ*such that*F*⊆̃*K*⊆̃*Cl*_{τ}_{ω}(*F*).*ω*(_{s}*X, τ, A*) will denote the family of all soft*ω*-open sets in (_{s}*X, τ, A*).### Theorem 2.2

Let (

*X, τ, A*) be a STS. Then*τ*⊆*ω*(_{s}*X, τ, A*) ⊆*SO*(*X, τ, A*).### Proof

Let

*G*∈*τ*, choose*F*=*G*. Then*F*∈*τ*and*F*⊆̃*G*⊆̃*Cl*_{τ}_{ω}(*F*) which implies that*G*∈*ω*(_{s}*X, τ, A*). This shows that*τ*⊆*ω*(_{s}*X, τ, A*). Let*K*∈*ω*(_{s}*X, τ, A*), then by definition we find*F*∈*τ*such that*F*⊆̃*K*⊆̃*Cl*_{τ}_{ω}(*F*) ⊆̃*Cl*(_{τ}*F*) which implies that*K*∈*SO*(*X, τ, A*). It follows that*ω*(_{s}*X, τ, A*) ⊆*SO*(*X, τ, A*).The following three examples show that none of the two inclusions in Theorem 2.2 is equality in general:

### Example 2.3

Let

*X*= ℝ and*A*= [0, 1]. Let*F*,*G*∈*SS*(*X, A*) such that for every*a*∈*A, F*(*a*) = ℕ and*G*(*a*) = ℚ. Let^{c}*τ*= {0, 1_{A},_{A}*F*,*G*,*F*∪̃*G*}. One can easily check that*Cl*_{τ}_{ω}(*F*) =*F*,*Cl*(_{τ}*F*) = 1–_{A}*G*and*Cl*_{τ}_{ω}(*G*) = 1–_{A}*F*. Hence 1–_{A}*G*∈*SO*(*X, τ, A*) –*ω*(_{s}*X, τ, A*) and 1–_{A}*F*∈*ω*(_{s}*X, τ, A*) –*τ*.### Example 2.4

Let

*X*= ℝ and*A*= ℕ. Let*τ*= {*F*∈*SS*(*X, A*) :*F*(*a*) belongs to the usual topology on ℝ}.Let

*G*∈*SS*(*X, A*) such that for every*a*∈*A, G*(*a*) = [0, 1). Then*G*∈*ω*(_{s}*X, τ, A*) –*τ*.### Example 2.5

Let

*X*= ℚ and*A*= ℝ. Let*F, G*∈*SS*(*X, A*) such that for every*a*∈*A, F*(*a*) = {1} and*G*(*a*) = {1, 2}. Let*τ*= {0, 1_{A},_{A}*F*}. Then*G*∈*SO*(*X, τ, A*) –*ω*(_{s}*X, τ, A*).### Theorem 2.6

If (

*X, τ, A*) is a soft anti-locally countable STS, then*ω*(_{s}*X, τ, A*) =*SO*(*X, τ, A*).### Proof

By Theorem 2.2, we only need to see that

*SO*(*X, τ, A*) ⊆*ω*(_{s}*X, τ, A*). Let*K*∈*SO*(*X, τ, A*), then there is*F*∈*τ*such that*F*⊆̃*K*⊆̃*Cl*(_{τ}*F*). By Proposition 1.4,*Cl*(_{τ}*F*) =*Cl*_{τ}_{ω}(*F*). Therefore,*K*∈*ω*(_{s}*X, τ, A*). This shows that*SO*(*X, τ, A*) ⊆*ω*(_{s}*X, τ, A*).### Theorem 2.7

For any STS (

*X, τ, A*),*SO*(*X, τ*_{ω}*, A*) =*ω*(_{s}*X, τ*_{ω}*, A*).### Proof

Let (

*X, τ, A*) be STS. By Theorem 2.2, we only need to show that*SO*(*X, τ*_{ω}*, A*) ⊆̃*ω*(_{s}*X, τ*_{ω}*, A*). Let*K*∈*SO*(*X, τ*_{ω}*, A*), then there is*F*∈*τ*such that_{ω}*F*⊆̃*K*⊆̃*Cl*_{(}_{τ}_{ω}_{)}_{ω}(*F*). As by Proposition 1.6, (*τ*)_{ω}=_{ω}*τ*, then_{ω}*Cl*_{(}_{τ}_{ω}_{)}_{ω}(*F*) =*Cl*_{τ}_{ω}(*F*). Hence,*K*∈*ω*(_{s}*X, τ*_{ω}*, A*). This ends the proof that*SO*(*X, τ*_{ω}*, A*) ⊆*ω*(_{s}*X, τ*_{ω}*, A*).### Theorem 2.8

If (

*X, τ, A*) is a soft locally countable STS, then*τ*=*ω*(_{s}*X, τ, A*).### Proof

By Theorem 2.2, we only need to see that

*ω*(_{s}*X, τ, A*) ⊆*τ*. Let*K*∈*ω*(_{s}*X, τ, A*). Then there is*F*∈*τ*such that*F*⊆̃*K*⊆̃*Cl*_{τ}_{ω}(*F*). By Proposition 1.5,*Cl*_{τ}_{ω}(*F*) =*F*which implies that*K*=*F*. Thus,*K*∈*τ*. This shows that*τ*=*ω*(_{s}*X, τ, A*).Soft

*ω*-open sets and soft*ω*-open sets are independent of each other:_{s}### Example 2.9

Let

*X*= ℝ and*A*= ℕ. Let*F, G, K*∈*SS*(*X, A*) defined by*F*(*a*) = {*a*} ∪ (2, 3),*G*(*a*) = [0, ∞) and*K*(*a*) = (2, 3) for all*a*∈*A*. Let*τ*= {0_{A}*,*1_{A}*, F*}. As (*X, τ, A*) is soft anti-locally countable, then by Proposition 1.5,*Cl*_{τ}_{ω}(*F*) =*Cl*(_{τ}*F*) = 1. So_{A}*G*∈*ω*(_{s}*X, τ, A*) –*τ*and_{ω}*K*∈*τ*–_{ω}*ω*(_{s}*X, τ, A*).### Theorem 2.10

Let (

*X, τ, A*) be a STS and let*K*∈*SS*(*X, A*). Then*K*is soft*ω*-open in (_{s}*X, τ, A*) if and only if*K*⊆̃*Cl*_{τ}_{ω}(*int*(_{τ}*K*)).### Proof

### Necessity

Suppose that

*K*is soft*ω*-open set in (_{s}*X, τ, A*). Then we find*F*∈*τ*such that*F*⊆̃*K*⊆̃*Cl*_{τ}_{ω}(*F*). As*F*⊆̃*K*, then*F*=*int*(_{τ}*F*) ⊆̃*int*(_{τ}*K*), which implies that*Cl*_{τ}_{ω}(*F*) ⊆̃*Cl*_{τ}_{ω}(*int*(_{τ}*K*)). It follows that*K*⊆̃*Cl*_{τ}_{ω}(*int*(_{τ}*K*)).### Sufficiency

Suppose that

*K*⊆̃*Cl*_{τ}_{ω}(*int*(_{τ}*K*)). Put*F*=*int*(_{τ}*K*). Then*F*∈*τ*with*F*⊆̃*K*⊆̃*Cl*_{τ}_{ω}(*int*(_{τ}*K*)). Hence,*K*is soft*ω*-open set in (_{s}*X, τ, A*).### Theorem 2.11

Let (

*X, τ, A*) be a STS. If {*K*:_{α}*α*∈ Δ} ⊆*ω*(_{s}*X, τ, A*), then$\tilde{\underset{\alpha \in \mathrm{\Delta}}{\cup}}{K}_{\alpha}\in {\omega}_{s}(X,\tau ,A)$ .### Proof

For every

*α*∈ Δ, we find*F*∈_{α}*τ*such that*F*⊆̃_{α}*K*⊆̃_{α}*Cl*_{τ}_{ω}(*F*). Then_{α}$\tilde{\underset{\alpha \in \mathrm{\Delta}}{\cup}}{F}_{\alpha}\in \tau $ and$\tilde{\underset{\alpha \in \mathrm{\Delta}}{\cup}}{F}_{\alpha}\tilde{\subseteq}\tilde{\underset{\alpha \in \mathrm{\Delta}}{\cup}}{K}_{\alpha}\tilde{\subseteq}\tilde{\underset{\alpha \in \mathrm{\Delta}}{\cup}}{Cl}_{{\tau}_{\omega}}({K}_{\alpha})\tilde{\subseteq}{Cl}_{{\tau}_{\omega}}\left(\tilde{\underset{\alpha \in \mathrm{\Delta}}{\cup}}{K}_{\alpha}\right)$ . Therefore,$\tilde{\underset{\alpha \in \mathrm{\Delta}}{\cup}}{K}_{\alpha}\in {\omega}_{s}(X,\tau ,A)$ .Soft intersection of two soft

*ω*-open sets is not a soft_{s}*ω*open in general:_{s}### Example 2.12

Let

*X*= ℝ and*A*= ℕ. Let*F, G*∈*SS*(*X*,*A*) defined by*F*(*a*) = (0, ∞) and*G*(*a*) = (–∞, 0) for all*a*∈*A*. Let*τ*= {0, 1_{A},_{A}*F*,*G*,*F*∪̃*G*}. As (*X, τ, A*) is soft anti-locally countable, then by Proposition 1.4,*Cl*_{τ}_{ω}(*F*) =*Cl*(_{τ}*F*) = 1–_{A}*G*and*Cl*_{τ}_{ω}(*G*) =*Cl*(_{τ}*G*) = 1–_{A}*F*. Then 1–_{A}*F*, 1–_{A}*G*∈*ω*(_{s}*X, τ, A*) but ((1–_{A}*F*) ∩̃ (1–_{A}*G*))(*a*) = {0} for all*a*∈*A*and hence (1–_{A}*F*) ∩̃ (1–_{A}*G*) ∉*ω*(_{s}*X, τ, A*).### Theorem 2.13

Let (

*X, τ, A*) be a STS. If*G*∈*τ*and*K*∈*ω*(_{s}*X, τ, A*), then*G*∩̃*K*∈*ω*(_{s}*X, τ, A*).### Proof

Let

*G*∈*τ*and*K*∈*ω*(_{s}*X, τ, A*). As*K*∈*ω*(_{s}*X, τ, A*), then there is*F*∈*τ*such that*F*⊆̃*K*⊆̃*Cl*_{τ}_{ω}(*F*). Then*G*∩̃*F*∈*τ*and*G*∩̃*F*⊆̃*G*∩̃*K*⊆̃*G*∩̃*Cl*_{τ}_{ω}(*F*) ⊆̃*Cl*_{τ}_{ω}(*G*∩̃*F*). This ends the proof that*G*∩̃*K*∈*ω*(_{s}*X, τ, A*).### Theorem 2.14

Let (

*X, τ, A*) be a STS. If*K*∈*ω*(_{s}*X, τ, A*) and*K*⊆̃*G*⊆̃*Cl*_{τ}_{ω}(*K*), then*G*∈*ω*(_{s}*X, τ, A*).### Proof

Suppose that

*K*∈*ω*(_{s}*X, τ, A*) and*K*⊆̃*G*⊆̃*Cl*_{τ}_{ω}(*K*). As*K*∈*ω*(_{s}*X, τ, A*), then there is*F*∈*τ*such that*F*⊆̃*K*⊆̃*Cl*_{τ}_{ω}(*F*). As*K*⊆̃*Cl*_{τ}_{ω}(*F*), then*Cl*_{τ}_{ω}(*K*) ⊆̃*Cl*_{τ}_{ω}(*F*). As*G*⊆̃*Cl*_{τ}_{ω}(*K*), then*G*⊆̃*Cl*_{τ}_{ω}(*F*). Thus, we have*F*∈*τ*and*F*⊆̃*K*⊆̃*G*⊆̃*Cl*_{τ}_{ω}(*K*) ⊆̃*Cl*_{τ}_{ω}(*F*).Therefore,

*G*∈*ω*(_{s}*X, τ, A*).### Theorem 2.15

For any STS (

*X, τ, A*),$$\tau =\{{int}_{\tau}(K):K\in {\omega}_{s}(X,\tau ,A)\}.$$

### Proof

As

*int*(_{τ}*K*) ∈*τ*for every*K*∈*ω*(_{s}*X, τ, A*), then {*int*(_{τ}*K*) :*K*∈*ω*(_{s}*X, τ, A*)} ⊆*τ*. Conversely, let*F*∈*τ*, then*F*=*int*(_{τ}*F*). On the other hand, by Theorem 2.2,*F*∈*ω*(_{s}*X, τ, A*). It follows that*τ*⊆ {*int*(_{τ}*K*) :*K*∈*ω*(_{s}*X, τ, A*)}.### Definition 2.16

Let (

*X, τ, A*) be a STS and let*M*∈*SS*(*X*,*A*). Then*M*is called soft*ω*-closed set in (_{s}*X, τ, A*) if 1–_{A}*M*is soft*ω*-open set in (_{s}*X, τ, A*).### Theorem 2.17

Let (

*X, τ, A*) be a STS. Then(a) Every soft closed set in (

*X, τ, A*) is soft*ω*-closed set in (_{s}*X, τ, A*).(b) Every soft

*ω*-closed set in (_{s}*X, τ, A*) is soft semi-closed set in (*X, τ, A*).(c) 0

and 1_{A}are soft_{A}*ω*-closed sets in (_{s}*X, τ, A*).### Proof

(a) Let

*M*be a soft closed set in (*X, τ, A*), then 1–_{A}*M*∈*τ*and by Theorem 2.2 1–_{A}*M*∈*ω*(_{s}*X, τ, A*). Thus,*M*is a soft*ω*-closed set in (_{s}*X, τ, A*).(b) Let

*M*be a soft*ω*-closed set in (_{s}*X, τ, A*), then 1–_{A}*M*∈*ω*-(_{s}*X, τ, A*) and by Theorem 2.2 1–_{A}*M*∈*SO*(*X, τ, A*). Thus,*M*is soft semi-closed set in (*X, τ, A*).(c) Follows by (a).

One can use Example 2.3 to show that none of the implications in Theorem 2.15 (a), (b) is reversible in general.

### Theorem 2.18

Let (

*X, τ, A*) be a STS. If*M*is soft_{α}*ω*-closed in (_{s}*X, τ, A*) for all*α*∈ Δ, then$\tilde{\underset{\alpha \in \mathrm{\Delta}}{\cap}}{M}_{\alpha}$ is soft*ω*_{s}*-*closed in (*X, τ, A*).### Proof

Suppose that

*M*is soft_{α}*ω*-closed in (_{s}*X, τ, A*) for all*α*∈ Δ. Then for every*α*∈ Δ, 1–_{A}*M*∈_{α}*ω*(_{s}*X, τ, A*). Thus, by Theorem 2.9, we have${1}_{A}-\left(\tilde{\underset{\alpha \in \mathrm{\Delta}}{\cap}}{M}_{\alpha}\right)=\tilde{\underset{\alpha \in \mathrm{\Delta}}{\cup}}({1}_{A}-{M}_{\alpha})\in {\omega}_{s}(X,\tau ,A)$ . It follows that$\tilde{\underset{\alpha \in \mathrm{\Delta}}{\cap}}{M}_{\alpha}$ is soft*ω*_{s}*-*closed in (*X, τ, A*).Soft

*ω*-closedness is not closed under finite soft union:_{s}### Example 2.19

Consider (

*X, τ, A*),*F*, and*G*as in Example 2.12. As 1–_{A}*F*, 1–_{A}*G*∈*ω*(_{s}*X, τ, A*), then*F*and*G*are soft*ω*-closed in (_{s}*X, τ, A*). As 1– (_{A}*F*∪̃*G*) = (1–_{A}*F*) ∩̃ (1–_{A}*G*) ∉*ω*(_{s}*X, τ, A*), then*F*∪̃*G*is not soft*ω*-closed in (_{s}*X, τ, A*).### Theorem 2.20

Let (

*X, τ, A*) be a STS. If*M*is soft closed in (*X, τ, A*) and*N*is soft*ω*-closed in (_{s}*X, τ, A*), then*M*∪̃*N*is soft*ω*-closed in (_{s}*X, τ, A*).### Proof

Suppose that

*M*is soft closed in (*X, τ, A*) and*N*is soft*ω*-closed in (_{s}*X, τ, A*). Then 1–_{A}*M*∈*τ*and 1–_{A}*N*∈*ω*(_{s}*X, τ, A*). So by Theorem 2.13, 1– (_{A}*M*∪̃*N*) = (1–_{A}*M*) ∩̃ (1–_{A}*N*) ∈*ω*(_{s}*X, τ, A*). It follows that*M*∪̃*N*is soft*ω*_{s}*-*closed in (*X, τ, A*).### Theorem 2.21

Let (

*X, τ, A*) be a STS and let*M*∈*SS*(*X, A*). Then*M*is soft*ω*-closed in (_{s}*X, τ, A*) if and only if$${int}_{{\tau}_{\omega}}({Cl}_{\tau}(M))\tilde{\subseteq}M.$$

### Proof

### Necessity

Assume that

*M*is soft*ω*-closed in (_{s}*X, τ, A*). Then 1–_{A}*M*is soft*ω*-open in (_{s}*X, τ, A*). Theorem Theorem 2.10 implies that 1–_{A}*M*⊆̃*Cl*_{τ}_{ω}(*int*(1_{τ}–_{A}*M*)), and so$$\begin{array}{l}{int}_{{\tau}_{\omega}}({Cl}_{\tau}\hspace{0.17em}(M))={Ext}_{{\tau}_{\omega}}\hspace{0.17em}({1}_{A}-{Cl}_{\tau}\hspace{0.17em}(M))\\ ={Ext}_{{\tau}_{\omega}}\hspace{0.17em}({Ext}_{\tau}\hspace{0.17em}(M))\\ ={1}_{A}-{Cl}_{{\tau}_{\omega}}\hspace{0.17em}({Ext}_{\tau}\hspace{0.17em}(M))\\ ={1}_{A}-{Cl}_{{\tau}_{\omega}}\hspace{0.17em}({int}_{\tau}\hspace{0.17em}({1}_{A}-M))\\ \tilde{\subseteq}M.\end{array}$$

### Sufficiency

Suppose that

*int*_{τ}_{ω}(*Cl*(_{τ}*M*)) ⊆̃*M*. Then$$\begin{array}{l}{1}_{A}-M\subseteq {1}_{A}-{int}_{{\tau}_{\omega}}({Cl}_{\tau}(M))\\ ={1}_{A}-{Ext}_{{\tau}_{\omega}}\hspace{0.17em}({1}_{A}-{Cl}_{\tau}\hspace{0.17em}(M))\\ ={1}_{A}-{Ext}_{{\tau}_{\omega}}\hspace{0.17em}({Ext}_{\tau}\hspace{0.17em}(M))\\ ={Cl}_{{\tau}_{\omega}}\hspace{0.17em}({Ext}_{\tau}\hspace{0.17em}(M))\\ ={Cl}_{{\tau}_{\omega}}\hspace{0.17em}({int}_{\tau}\hspace{0.17em}({1}_{A}-M)).\end{array}$$

Therefore, by Theorem 2.10, 1

–_{A}*M*is soft*ω*-open in (_{s}*X, τ, A*) and so*M*is soft*ω*-closed in (_{s}*X, τ, A*).### Theorem 2.22

If

*f*: (_{pu}*X, τ, A*) → (*Y, σ, B*) is a soft open mapping such that*f*: (_{pu}*X, τ*_{ω}*, A*) → (*Y, σ*_{ω}*, B*) is soft continuous, then for every*K*∈*ω*(_{s}*X, τ, A*), we have*f*(_{pu}*K*) ∈*ω*(_{s}*Y, σ, B*).### Proof

Let

*K*∈*ω*(_{s}*X, τ, A*). Then there exists*F*∈*τ*such that*F*⊆̃*K*⊆̃*Cl*_{τ}_{ω}(*F*). Thus we have*f*(_{pu}*F*) ⊆̃*f*(_{pu}*K*) ⊆̃*f*(_{pu}*Cl*_{τ}_{ω}(*F*)). As*f*: (_{pu}*X, τ, A*) → (*Y, σ, B*) is soft open mapping, then*f*(_{pu}*F*) ∈*σ*. As*f*: (_{pu}*X, τ*_{ω}*, A*) → (*Y, σ*_{ω}*, B*) is soft continuous, then*f*(_{pu}*Cl*_{τ}_{ω}(*F*)) ⊆̃*Cl*_{τ}_{ω}(*f*(_{pu}*F*)). Therefore,*f*(_{pu}*K*) ∈*ω*(_{s}*Y, σ, B*).In Theorem 2.22, we cannot drop the condition ‘soft open mapping’:

### Example 2.23

Let

*X*= ℝ,*A*= ℕ,*F, G*∈*SS*(*X, A*) with*F*(*a*) = ℝ and*G*(*a*) = {0} for all*a*∈*A*,*τ*=*SS*(*X, A*), and*σ*= {0_{B}*,*1_{B}*, F*}. Define*p*:*X*→*X*and*u*:*A*→*A*by*p*(*x*) = 0 for all*x*∈*X*and*u*(*a*) =*a*for all*a*∈*A*. Then clearly that*f*: (_{pu}*X, τ*_{ω}*, A*) → (*X, τ*_{ω}*, A*) is soft continuous, but*G*∈*ω*(_{s}*X, τ*_{ω}*, A*) while*f*(_{pu}*G*) =*G*∉*ω*(_{s}*X, τ*_{ω}*, A*).

- 3. Soft
ω -Closure and Soft_{s}ω -Interior_{s} ### Definition 3.1

Let (

*X, τ, A*) be a STS and*H*∈*SS*(*X, A*). The soft*ω*-closure of_{s}*H*in (*X, τ, A*) is denoted by*ω*-_{s}*Cl*(_{τ}*H*) and defined by*ω*-_{s}*Cl*(_{τ}*H*) = ∩̃ {*M*:*M*is soft*ω*-closed in (_{s}*X, τ, A*) and*H*⊆̃ }.### Theorem 3.2

Let (

*X, τ, A*) be a STS and*H*∈*SS*(*X, A*). Then(a)

*ω*-_{s}*Cl*(_{τ}*H*) is the smallest soft*ω*-closed in (_{s}*X, τ, A*) containing*H*.(b)

*H*is soft*ω*-closed in (_{s}*X, τ, A*) if and only if*H*=*ω*-_{s}*Cl*(_{τ}*H*).### Proof

(a) Follows from Definition 3.1 and Theorem 2.18.

(b) Follows immediately by (a).

### Theorem 3.3

Let (

*X, τ, A*) be a STS and*H*∈*SS*(*X, A*). Then*H*⊆̃*S*-*Cl*(_{τ}*H*) ⊆̃*ω*-_{s}*Cl*(_{τ}*H*) ⊆̃*Cl*(_{τ}*H*).### Proof

Follows from the definitions and parts (a) and (b) of Theorem 2.17.

In Theorem 3.3

*S*-*Cl*(_{τ}*H*) ≠*ω*-_{s}*Cl*(_{τ}*H*) and*ω*-_{s}*Cl*(_{τ}*H*) ≠*Cl*(_{τ}*H*), in general:### Example 3.4

Consider Example 2.3. We proved that 1

–_{A}*G*∈*SO*(*X, τ, A*) –*ω*(_{s}*X, τ, A*) and 1–_{A}*F*∈*ω*(_{s}*X, τ, A*) –*τ*. So, we have*F*as an*ω*-closed set in (_{s}*X, τ, A*) that is not closed set in (*X, τ, A*), and*G*as a semi-closed set in (*X, τ, A*) that is not*ω*-closed set in (_{s}*X, τ, A*). Thus, by Proposition 3.9 (2) of [26] and Theorem 3.2 (b), *S*-*Cl*(_{τ}*G*) =*G*≠*ω*-_{s}*Cl*(_{τ}*H*) and*ω*-_{s}*Cl*(_{τ}*F*) =*F*≠*Cl*(_{τ}*F*).### Theorem 3.5

Let (

*X, τ, A*) be a STS and*H*∈*SS*(*X, A*). Then(a) If

*H*is soft closed in (*X, τ, A*), then*H*=*S*-*Cl*(_{τ}*H*) =*ω*-_{s}*Cl*(_{τ}*H*) =*Cl*(_{τ}*H*).(b) If

*H*is soft*ω*-closed in (_{s}*X, τ, A*), then*H*=*S*-*Cl*(_{τ}*H*) =*ω*-_{s}*Cl*(_{τ}*H*).### Proof

(a) Suppose that soft

*H*is closed in (*X, τ, A*). Then*H*=*Cl*(_{τ}*H*) and by Theorem 3.3 we have*H*=*S*-*Cl*(_{τ}*H*) =*ω*-_{s}*Cl*(_{τ}*H*) =*Cl*(_{τ}*H*).(b) Suppose that

*H*is soft*ω*-closed in (_{s}*X, τ, A*). Then by Theorem 3.2 (b) we have*H*=*ω*-_{s}*Cl*(_{τ}*H*), and by Theorem 3.3 we conclude that*H*=*S*-*Cl*(_{τ}*H*) =*ω*-_{s}*Cl*(_{τ}*H*).### Theorem 3.6

If (

*X, τ, A*) is a soft anti-locally countable STS then for every*H*∈*SS*(*X, A*),*S*-*Cl*(_{τ}*H*) =*ω*-_{s}*Cl*(_{τ}*H*).### Proof

As (

*X, τ, A*) is soft anti-locally countable, then by Theorem 2.6 we have*ω*(_{s}*X, τ, A*) =*SO*(*X, τ, A*) and hence*S*-*Cl*(_{τ}*H*) =*ω*-_{s}*Cl*(_{τ}*H*).### Theorem 3.7

If (

*X, τ, A*) is a soft locally countable STS then for every*H*∈*SS*(*X, A*),*ω*-_{s}*Cl*(_{τ}*H*) =*Cl*(_{τ}*H*).### Proof

As (

*X, τ, A*) is soft locally countable, then by Theorem 2.8 we have*τ*=*ω*(_{s}*X, τ, A*) and hence*ω*-_{s}*Cl*(_{τ}*H*) =*Cl*(_{τ}*H*).### Theorem 3.8

If (

*X, τ, A*) is a STS and*H*∈*SS*(*X, A*), then*S*-*Cl*_{τ}_{ω}(*H*) =*ω*-_{s}*Cl*_{τ}_{ω}(*H*).### Proof

By Theorem 2.7 we have

*SO*(*X, τ*_{ω}*, A*) =*ω*(_{s}*X, τ*,_{ω}*A*) and hence*S*-*Cl*_{τ}_{ω}(*H*) =*ω*-_{s}*Cl*_{τ}_{ω}(*H*).### Theorem 3.9

Let (

*X, τ, A*) be a STS and*H, K*∈*SS*(*X, A*). Then(a) If

*H*⊆̃*K*, then*ω*-_{s}*Cl*(_{τ}*H*) ⊆̃*ω*-_{s}*Cl*(_{τ}*K*).(b)

*ω*-_{s}*Cl*(_{τ}*H*) ∪̃*ω*-_{s}*Cl*(_{τ}*K*) ⊆̃*ω*-_{s}*Cl*(_{τ}*H*∪̃*K*).(c)

*ω*-_{s}*Cl*(_{τ}*H*∩̃*K*) ⊆̃*ω*-_{s}*Cl*(_{τ}*H*) ∩̃*ω*-_{s}*Cl*(_{τ}*K*).### Proof

(a) Suppose that

*H*⊆̃*K*. As*K*⊆̃*ω*-_{s}*Cl*(_{τ}*K*), then*H*⊆̃*ω*-_{s}*Cl*(_{τ}*K*). By Theorem 3.2 (a),*ω*-_{s}*Cl*(_{τ}*K*) is soft*ω*closed in (_{s}*X, τ, A*). Again by Theorem 3.2 (a),*ω*-_{s}*Cl*(_{τ}*H*) ⊆̃*ω*-_{s}*Cl*(_{τ}*K*).(b) As

*H*⊆̃*H*∪̃*K*and*K*⊆̃*H*∪̃*K*, then by (a),*ω*-_{s}*Cl*(_{τ}*H*) ⊆̃*ω*-_{s}*Cl*(_{τ}*H*∪̃*K*) and*ω*-_{s}*Cl*(_{τ}*K*) ⊆̃*ω*-_{s}*Cl*(_{τ}*H*∪̃*K*). Hence*ω*-_{s}*Cl*(_{τ}*H*) ∪̃*ω*-_{s}*Cl*(_{τ}*K*) ⊆̃*ω*-_{s}*Cl*(_{τ}*H*∪̃*K*).(c) As

*H*∩̃*K*⊆̃*H*and*H*∩̃*K*⊆̃*K*, then by (a),*ω*-_{s}*Cl*(_{τ}*H*∩̃*K*) ⊆̃*ω*-_{s}*Cl*(_{τ}*H*) and*ω*-_{s}*Cl*(_{τ}*H*∩̃*K*) ⊆̃*ω*-_{s}*Cl*(_{τ}*K*). Hence*ω*-_{s}*Cl*(_{τ}*H*∩̃*K*) ⊆̃*ω*-_{s}*Cl*(_{τ}*H*) ∩̃*ω*-_{s}*Cl*(_{τ}*K*).The soft inclusion in Theorem 3.9 (b) cannot be replaced by soft equality in general:

### Example 3.10

Consider (

*X, τ, A*) as in Example 2.19. We proved that*F*and*G*are soft*ω*-closed in (_{s}*X, τ, A*) while*F*∪̃*G*is not soft*ω*-closed in (_{s}*X, τ, A*). Therefore, by Theorem 3.2 (b),*ω*-_{s}*Cl*(_{τ}*F*) ∪̃*ω*-_{s}*Cl*(_{τ}*G*) =*F*∪̃*G*while*ω*-_{s}*Cl*(_{τ}*F*∪̃*G*) ≠*F*∪̃*G*.### Theorem 3.11

If (

*X, τ, A*) is a soft locally countable STS, then for any*H, K*∈*SS*(*X, A*) we have*ω*-_{s}*Cl*(_{τ}*H*) ∪̃*ω*-_{s}*Cl*(_{τ}*K*) =*ω*-_{s}*Cl*(_{τ}*H*∪̃*K*).### Proof

As (

*X, τ, A*) is soft locally countable, then by Theorem 3.7,$$\begin{array}{l}{\omega}_{s}-{Cl}_{\tau}\hspace{0.17em}(H\tilde{\cup}K)={Cl}_{\tau}\hspace{0.17em}(H\tilde{\cup}K)\\ ={Cl}_{\tau}\hspace{0.17em}(H)\tilde{\cup}{Cl}_{\tau}\hspace{0.17em}(K)\\ ={\omega}_{s}-{Cl}_{\tau}(H)\tilde{\cup}{\omega}_{s}-{Cl}_{\tau}(K).\end{array}$$

### Theorem 3.12

Let (

*X, τ, A*) be a STS and let*H, K*∈*SS*(*X, A*). If*H*is soft closed in (*X, τ, A*), then*ω*-_{s}*Cl*(_{τ}*H*) ∪̃*ω*-_{s}*Cl*(_{τ}*K*) =*ω*-_{s}*Cl*(_{τ}*H*∪̃*K*).### Proof

By Theorem 3.9 (b) we only need to show that

*ω*-_{s}*Cl*(_{τ}*H*∪̃*K*) ⊆̃*ω*-_{s}*Cl*(_{τ}*H*) ∪̃*ω*-_{s}*Cl*(_{τ}*K*). As*H*is soft closed in (*X, τ, A*), then by Theorem 3.5 (a),*H*=*ω*-_{s}*Cl*(_{τ}*H*). So by Theorem 2.20,*ω*-_{s}*Cl*(_{τ}*H*) ∪̃*ω*-_{s}*Cl*(_{τ}*K*) is*ω*-closed in (_{s}*X, τ, A*). As*H*∪̃*K*⊆̃*ω*-_{s}*Cl*(_{τ}*H*) ∪̃*ω*-_{s}*Cl*(_{τ}*K*), then by Theorem 3.2 (a) we have*ω*-_{s}*Cl*(_{τ}*H*∪̃*K*) ⊆̃*ω*-_{s}*Cl*(_{τ}*H*) ∪̃*ω*-_{s}*Cl*(_{τ}*K*).The soft inclusion in Theorem 3.9 (c) cannot be replaced by soft equality in general:

### Example 3.13

Let

*X*= ℤ*×*ℤ,*A*= ℝ and*τ*= {*F*∈*SS*(*X, A*) :*F*(*a*) ∈*τ*for every_{cof}*a*∈*A*}. Let*H, K*∈*SS*(*X, A*) defined by*H*(*a*) = ℤ*×*{0} and*K*(*a*) = {0}*×*ℤ for all*a*∈*A*. Then (*H*∩̃*K*) (*a*) = {(0, 0)} for all*a*∈*A*. Note that*M*∈*SS*(*X, A*) is soft closed in (*X, τ, A*) if and only if for all*a*∈*A*, either*M*(*a*) is finite or*M*(*a*) = ℤ*×*ℤ. Then*Cl*(_{τ}*H*) =*Cl*(_{τ}*K*) = 1and_{A}*Cl*(_{τ}*H*∩̃*K*) =*H*∩̃*K*. As (*X, τ, A*) is soft locally countable, then by Theorem 3.7,*ω*-_{s}*Cl*(_{τ}*H*) =*Cl*(_{τ}*H*) = 1,_{A}*ω*-_{s}*Cl*(_{τ}*K*) =*Cl*(_{τ}*K*) = 1, and_{A}*ω*-_{s}*Cl*(_{τ}*H*∩̃*K*) =*H*∩̃*K*. Thus,*ω*-_{s}*Cl*(_{τ}*H*) ∩̃*ω*-_{s}*Cl*(_{τ}*K*) = 1while_{A}*ω*-_{s}*Cl*(_{τ}*H*∩̃*K*)*ω*-_{s}*Cl*(_{τ}*H*∩̃*K*) ≠ 1._{A}### Theorem 3.14

Let (

*X, τ, A*) be a STS and let*H*∈*SS*(*X*,*A*). Then*S*-*Cl*(_{τ}*Cl*(_{τ}*H*)) =*Cl*(_{τ}*S*-*Cl*(_{τ}*H*)) =*ω*-_{s}*Cl*(_{τ}*Cl*(_{τ}*H*)) =*Cl*(_{τ}*ω*-_{s}*Cl*(_{τ}*H*)) =*Cl*(_{τ}*H*).### Proof

As

*Cl*(_{τ}*H*) is soft closed in (*X, τ, A*), then, by Theorem 3.5 (a) we have*Cl*(_{τ}*H*) =*S*-*Cl*(_{τ}*Cl*(_{τ}*H*)) =*ω*-_{s}*Cl*(_{τ}*Cl*(_{τ}*H*)).By Theorem 3.3,

*H*⊆̃*S*-*Cl*(_{τ}*H*) ⊆̃*ω*-_{s}*Cl*(_{τ}*H*) ⊆̃*Cl*(_{τ}*H*). So,*Cl*(_{τ}*H*) ⊆̃*Cl*(_{τ}*S*-*Cl*(_{τ}*H*)) ⊆̃*Cl*(_{τ}*ω*-_{s}*Cl*(_{τ}*H*)) ⊆̃*Cl*(_{τ}*Cl*(_{τ}*H*)) =*Cl*(_{τ}*H*). Hence, we have*Cl*(_{τ}*H*) =*Cl*(_{τ}*S*-*Cl*(_{τ}*H*)) =*Cl*(_{τ}*ω*-_{s}*Cl*(_{τ}*H*)).### Theorem 3.15

Let (

*X, τ, A*) be a STS and let*H*∈*SS*(*X*,*A*). Then*ω*-_{s}*Cl*(_{τ}*S*-*Cl*(_{τ}*H*)) =*S*-*Cl*(_{τ}*ω*-_{s}*Cl*(_{τ}*H*)) =*ω*-_{s}*Cl*(_{τ}*H*).### Proof

As

*ω*-_{s}*Cl*(_{τ}*H*) is soft*ω*-closed in (_{s}*X, τ, A*), then by Theorem 3.5 (b) we have*ω*-_{s}*Cl*(_{τ}*H*) =*S*-*Cl*(_{τ}*ω*-_{s}*Cl*(_{τ}*H*)).By Theorem 3.3,

*H*⊆̃*S*-*Cl*(_{τ}*H*) ⊆̃*ω*-_{s}*Cl*(_{τ}*H*). So by Theorem 3.9 (a),*ω*-_{s}*Cl*(_{τ}*H*) ⊆̃*ω*-_{s}*Cl*(_{τ}*S*-*Cl*(_{τ}*H*)) ⊆̃*ω*-_{s}*Cl*(_{τ}*ω*-_{s}*Cl*(_{τ}*H*)) =*ω*-_{s}*Cl*(_{τ}*H*). Therefore,$${\omega}_{s}-{Cl}_{\tau}\hspace{0.17em}(S-{Cl}_{\tau}(H))={\omega}_{s}-{Cl}_{\tau}(H).$$

### Theorem 3.16

Let (

*X, τ, A*) be a STS and let*H*∈*SS*(*X*,*A*). Then*a*∊̃_{x}*ω*-_{s}*Cl*(_{τ}*H*) if and only if for all*K*∈*ω*(_{s}*X, τ, A*) with*a*∊̃_{x}*K*we have*K*∩̃*H*≠ 0._{A}### Proof

### Necessity

Suppose that

*a*∊̃_{x}*ω*-_{s}*Cl*(_{τ}*H*) and suppose to the contrary that there is*K*∈*ω*(_{s}*X, τ, A*) such that*a*∊̃_{x}*K*and*K*∩̃*H*= 0. So we have_{A}*H*⊆̃ 1–_{A}*K*with 1–_{A}*K*is soft*ω*-closed set in (_{s}*X, τ, A*), and hence*ω*-_{s}*Cl*(_{τ}*H*) ⊆̃ 1–_{A}*K*. As*a*∊̃_{x}*ω*-_{s}*Cl*(_{τ}*H*), then*a*∊̃ 1_{x}–_{A}*K*, a contradiction.### Sufficiency

By contradiction. Then we have

*a*∊̃ (1_{x}–_{A}*ω*-_{s}*Cl*(_{τ}*H*)) ∈*ω*(_{s}*X, τ, A*). Then by assumption (1–_{A}*ω*-_{s}*Cl*(_{τ}*H*)) ∩̃*H*= 0, a contradiction._{A}### Definition 3.17

Let (

*X, τ, A*) be a STS and*H*∈*SS*(*X, A*). The soft*ω*-interior of_{s}*H*in (*X, τ, A*) is denoted by*ω*-_{s}*int*(_{τ}*H*) and defined by*ω*-_{s}*int*(_{τ}*H*) = ∪̃ {*K*:*K*is soft*ω*-open in (_{s}*X, τ, A*) and*K*⊆̃*H*}.### Theorem 3.18

Let (

*X, τ, A*) be a STS and*H*∈*SS*(*X, A*). Then(a)

*ω*-_{s}*int*(_{τ}*H*) is the largest soft*ω*-open in (_{s}*X, τ, A*) contained in*H*.(b)

*H*is soft*ω*-open in (_{s}*X, τ, A*) if and only if*H*=*ω*_{s}*int*(_{τ}*H*).### Proof

(a) Follows from Definition 3.17 and Theorem 2.11.

(b) Follows immediately by (a).

### Theorem 3.19

Let (

*X, τ, A*) be a STS and*H*∈*SS*(*X, A*). Then*int*(_{τ}*H*) ⊆̃*ω*-_{s}*int*(_{τ}*H*) ⊆̃*S*-*int*(_{τ}*H*) ⊆̃*H*.### Proof

Follows from the definitions and Theorem 2.2.

In Theorem 3.19

*int*(_{τ}*H*) ≠*ω*-_{s}*int*(_{τ}*H*) and*ω*-_{s}*int*(_{τ}*H*) ≠*S*-*int*(_{τ}*H*), in general:### Example 3.20

Consider Example 2.3. We proved that 1

–_{A}*G*∈*SO*(*X, τ, A*) –*ω*(_{s}*X, τ, A*) and 1–_{A}*F*∈*ω*(_{s}*X, τ, A*) –*τ*. Thus, by Proposition 3.9 (2) and Theorem 3.2 (b),*S*-*int*(1_{τ}–_{A}*G*) = 1–_{A}*G*≠*ω*-_{s}*int*(1_{τ}–_{A}*G*) and*ω*-_{s}*int*(1_{τ}–_{A}*F*) = 1–_{A}*F*≠*int*(1_{τ}–_{A}*F*).### Theorem 3.21

Let (

*X, τ, A*) be a STS and*H*∈*SS*(*X, A*). Then(a) If

*H*is soft open in (*X, τ, A*), then*H*=*S*-*int*(_{τ}*H*) =*ω*-_{s}*int*(_{τ}*H*) =*int*(_{τ}*H*).(b) If

*H*is soft*ω*-open in (_{s}*X, τ, A*), then*H*=*S*-*int*(_{τ}*H*) =*ω*-_{s}*int*(_{τ}*H*).### Proof

(a) Suppose that

*H*is soft open in (*X, τ, A*). Then*H*=*int*(_{τ}*H*) and by Theorem 3.19 we have*H*=*S*-*int*(_{τ}*H*) =*ω*-_{s}*int*(_{τ}*H*) =*int*(_{τ}*H*).(b) Suppose that

*H*is soft*ω*-open in (_{s}*X, τ, A*). Then by Theorem 3.2 (b),*H*=*ω*-_{s}*int*(_{τ}*H*). Thus, by Theorem 3.3 we*ω*-_{s}*int*(_{τ}*H*) ⊆̃*S*-*int*(_{τ}*H*) ⊆̃*ω*-_{s}*int*(_{τ}*H*). Therefore,*H*=*S*-*int*(_{τ}*H*) =*ω*-_{s}*int*(_{τ}*H*).### Theorem 3.22

If (

*X, τ, A*) is a soft anti-locally countable STS then for every*H*∈*SS*(*X, A*),*S*-*int*(_{τ}*H*) =*ω*_{s}*int*(_{τ}*H*).### Proof

As (

*X, τ, A*) is soft anti-locally countable, then by Theorem 2.6 we have*ω*(_{s}*X, τ, A*) =*SO*(*X, τ, A*) and hence*S*-*int*(_{τ}*H*) =*ω*-_{s}*int*(_{τ}*H*).### Theorem 3.23

If (

*X, τ, A*) is a soft locally countable STS then for every*H*∈*SS*(*X, A*),*ω*-_{s}*int*(_{τ}*H*) =*int*(_{τ}*H*).### Proof

As (

*X, τ, A*) is soft locally countable, then by Theorem 2.8 we have*τ*=*ω*(_{s}*X, τ, A*) and hence*ω*-_{s}*int*(_{τ}*H*) =*int*(_{τ}*H*).### Theorem 3.24

If (

*X, τ, A*) is a STS and*H*∈*SS*(*X, A*), then*S*-*int*_{τ}_{ω}(*H*) =*ω*-_{s}*int*_{τ}_{ω}(*H*).### Proof

By Theorem 2.7 we have

*SO*(*X, τ*_{ω}*, A*) =*ω*(_{s}*X, τ*,_{ω}*A*) and hence*S*-*int*_{τ}_{ω}(*H*) =*ω*-_{s}*int*_{τ}_{ω}(*H*).### Theorem 3.25

Let (

*X, τ, A*) be a STS and*H, K*∈*SS*(*X, A*). Then(a) If

*H*⊆̃*K*, then*ω*-_{s}*int*(_{τ}*H*) ⊆̃*ω*-_{s}*int*(_{τ}*K*).(b)

*ω*-_{s}*int*(_{τ}*H*) ∪̃*ω*-_{s}*int*(_{τ}*K*) ⊆̃*ω*-_{s}*int*(_{τ}*H*∪̃*K*).(c)

*ω*-_{s}*int*(_{τ}*H*∩̃*K*) ⊆̃*ω*-_{s}*int*(_{τ}*H*) ∩̃*ω*-_{s}*int*(_{τ}*K*).### Proof

(a) Suppose that

*H*⊆̃*K*. Then we have*ω*-_{s}*int*(_{τ}*H*) ⊆̃*H*⊆̃*K*. By Theorem 3.18 (a),*ω*-_{s}*int*(_{τ}*H*) is soft*ω*-open in (_{s}*X, τ, A*). Again by Theorem 3.18 (a),*ω*-_{s}*int*(_{τ}*H*) ⊆̃*ω*_{s}*-**int*(_{τ}*K*).(b) As

*H*⊆̃*H*∪̃*K*and*K*⊆̃*H*∪̃*K*, then by (a),*ω*_{s}*-**int*(_{τ}*H*) ⊆̃*ω*-_{s}*int*(_{τ}*H*∪̃*K*) and*ω*-_{s}*int*(_{τ}*K*) ⊆̃*ω*-_{s}*int*(_{τ}*H*∪̃*K*). Hence*ω*-_{s}*int*(_{τ}*H*) ∪̃*ω*-_{s}*int*(_{τ}*K*) ⊆̃*ω*-_{s}*int*(_{τ}*H*∪̃*K*).(c) As

*H*∩̃*K*⊆̃*H*and*H*∩̃*K*⊆̃*K*, then by (a),*ω*_{s}*-int*(_{τ}*H*∩̃*K*) ⊆̃*ω*-_{s}*int*(_{τ}*H*) and*ω*-_{s}*int*(_{τ}*H*∩̃*K*) ⊆̃*ω*-_{s}*int*(_{τ}*K*). Hence*ω*-_{s}*int*(_{τ}*H*∩̃*K*) ⊆̃*ω*-_{s}*int*(_{τ}*H*) ∩̃*ω*-_{s}*int*(_{τ}*K*).The soft inclusion in Theorem 3.26 (c) cannot be replaced by soft equality in general:

### Example 3.27

Consider (

*X, τ, A*) as in Example 2.12. We proved that 1–_{A}*F*and 1–_{A}*G*are soft*ω*-open in (_{s}*X, τ, A*) while (1–_{A}*F*) ∩̃ (1–_{A}*G*) is not soft*ω*-open in (_{s}*X, τ, A*). Therefore, by Theorem 3.18 (b),$${\omega}_{s}-{int}_{\tau}({1}_{A}-F)\tilde{\cap}{\omega}_{s}-{int}_{\tau}({1}_{A}-G)=({1}_{A}-F)\tilde{\cap}({1}_{A}-G),$$

while

$${\omega}_{s}-{int}_{\tau}(({1}_{A}-F)\tilde{\cap}({1}_{A}-G)\ne ({1}_{A}-F)\tilde{\cap}({1}_{A}-G).$$

### Theorem 3.28

If (

*X, τ, A*) is a soft locally countable STS, then for any*H, K*∈*SS*(*X, A*) we have*ω*-_{s}*int*(_{τ}*H*) ∩̃*ω*_{s}*-int*(_{τ}*K*) =*ω*-_{s}*int*(_{τ}*H*∩̃*K*).### Proof

As (

*X, τ, A*) is soft locally countable, then by Theorem 3.23,$$\begin{array}{l}{\omega}_{s}-{int}_{\tau}\hspace{0.17em}(H\tilde{\cap}K)={int}_{\tau}\hspace{0.17em}(H\tilde{\cap}K)\\ ={Cl}_{\tau}\hspace{0.17em}(H)\tilde{\cap}{Cl}_{\tau}\hspace{0.17em}(K)\\ ={\omega}_{s}-{Cl}_{\tau}(H)\tilde{\cap}{\omega}_{s}-{Cl}_{\tau}(K).\end{array}$$

### Theorem 3.29

Let (

*X, τ, A*) be a STS and let*H, K*∈*SS*(*X, A*). If*H*is soft open in (*X, τ, A*), then*ω*-_{s}*int*(_{τ}*H*) ∩̃*ω*-_{s}*int*(_{τ}*K*) =*ω*-_{s}*int*(_{τ}*H*∩̃*K*).### Proof

By Theorem 3.25 (c) we only need to show that

*ω*_{s}*-int*(_{τ}*H*) ∩̃*ω*-_{s}*int*(_{τ}*K*) ⊆̃*ω*-_{s}*int*(_{τ}*H*∩̃*K*). As*H*is soft open in (*X, τ, A*), then by Theorem 3.18 (b),*H*=*ω*-_{s}*int*(_{τ}*H*). So by Theorem 2.13,*ω*-_{s}*int*(_{τ}*H*) ∩̃*ω*-_{s}*int*(_{τ}*K*) is soft*ω*-open in (_{s}*X, τ, A*). As*ω*-_{s}*int*(_{τ}*H*) ∩̃*ω*-_{s}*int*(_{τ}*K*) ⊆̃*H*∩̃*K*, then by Theorem 3.18 (a) we have*ω*-_{s}*int*(_{τ}*H*) ∩̃*ω*-_{s}*int*(_{τ}*K*) ⊆̃*ω*-_{s}*int*(_{τ}*H*∩̃*K*).The soft inclusion in Theorem 3.25 (b) cannot be replaced by soft equality in general:

### Example 3.30

Consider (

*X, τ, A*) in Example 3.13 and let*B*= {–*n*:*n*∈ ℕ}. Let*H, K*∈*SS*(*X, A*) defined by*H*(*a*) = ℤ*×*ℕ and*K*(*a*) = ({0} ∪*B*)*×*ℤ for all*a*∈*A*. As (*X, τ, A*) is soft locally countable, then by Theorem 3.23,*ω*-_{s}*int*(_{τ}*H*) =*int*(_{τ}*H*) = 0,_{A}*ω*-_{s}*int*(_{τ}*K*) =*int*(_{τ}*K*) = 0, and_{A}*ω*-_{s}*Cl*(_{τ}*H*∪̃*K*) =*int*(_{τ}*H*∪̃*K*) =*int*(1_{τ}) = 1_{A}. Thus,_{A}*ω*-_{s}*int*(_{τ}*H*) ∪̃*ω*-_{s}*int*(_{τ}*K*) = 0while_{A}*ω*-_{s}*Cl*(_{τ}*H*∪̃*K*) ≠ 0._{A}### Theorem 3.31

Let (

*X, τ, A*) be a STS and let*H*∈*SS*(*X, A*). Then*int*(_{τ}*H*) =*ω*-_{s}*int*(_{τ}*int*(_{τ}*H*)) =*S*-*int*(_{τ}*int*(_{τ}*H*)) =*int*(_{τ}*ω*-_{s}*int*(_{τ}*H*)) =*int*(_{τ}*S*-*int*(_{τ}*H*)).### Proof

As

*int*(_{τ}*H*) is soft open in (*X, τ, A*), then by Theorem 3.21 (a) we have*int*(_{τ}*H*) =*S*-*int*(_{τ}*int*(_{τ}*H*)) =*ω*_{s}*-int*(_{τ}*int*(_{τ}*H*)).By Theorem 3.19 we have

*int*(_{τ}*H*) ⊆̃*ω*-_{s}*int*(_{τ}*H*) ⊆̃*S-int*(_{τ}*H*) ⊆̃*H*. So,*int*(_{τ}*H*) =*int*(_{τ}*int*(_{τ}*H*)) ⊆̃*int*(_{τ}*ω*-_{s}*int*(_{τ}*H*)) ⊆̃*int*(_{τ}*S*-*int*(_{τ}*H*)) ⊆̃*int*(_{τ}*H*).Hence,

$${int}_{\tau}(H)={int}_{\tau}\hspace{0.17em}({\omega}_{s}-{int}_{\tau}(H))={int}_{\tau}\hspace{0.17em}(S-{int}_{\tau}(H)).$$

### Theorem 3.32

Let (

*X, τ, A*) be a STS and let*H*∈*SS*(*X, A*). Then*ω*-_{s}*int*(_{τ}*S*-*int*(_{τ}*H*)) =*S*-*int*(_{τ}*ω*-_{s}*int*(_{τ}*H*)) =*ω*_{s}*-**int*(_{τ}*H*).### Proof

As

*ω*-_{s}*int*(_{τ}*H*) is soft*ω*-open in (_{s}*X, τ, A*), then by Theorem 3.21 (b) we have*ω*-_{s}*int*(_{τ}*H*) =*S*-*int*(_{τ}*ω*-_{s}*int*(_{τ}*H*))By Theorem 3.19 we have

$${\omega}_{s}-{int}_{\tau}(H)\tilde{\subseteq}S-{int}_{\tau}(H)\tilde{\subseteq}H.$$

So by Theorem 3.25 (a) we have

*ω*-_{s}*int*(_{τ}*H*) =*ω*-_{s}*int*(_{τ}*ω*_{s}*-**int*(_{τ}*H*)) ⊆̃*ω*-_{s}*int*(_{τ}*S*-*int*(_{τ}*H*)) ⊆̃*ω*-_{s}*int*(_{τ}*H*). Therefore,*ω*-_{s}*int*(_{τ}*H*) =*ω*-_{s}*int*(_{τ}*S*-*int*(_{τ}*H*)).### Theorem 3.33

Let (

*X, τ, A*) be a STS and let*H*∈*SS*(*X, A*). Then(a)

*ω*-_{s}*int*(1_{τ}–_{A}*H*) ∩̃*ω*-_{s}*Cl*(_{τ}*H*) = 0._{A}(b) 1

=_{A}*ω*-_{s}*int*(1_{τ}–_{A}*H*) ∪̃*ω*-_{s}*Cl*(_{τ}*H*).(c) 1

–_{A}*ω*-_{s}*Cl*(_{τ}*H*) =*ω*-_{s}*int*(1_{τ}–_{A}*H*) and 1–_{A}*ω*_{s}*-**int*(1_{τ}–_{A}*H*) =*ω*-_{s}*Cl*(_{τ}*H*).### Proof

(a) Suppose to the contrary that there exists

*a*∊̃ (_{x}*ω*-_{s}*int*(1_{τ}–_{A}*H*) ∩̃*ω*-_{s}*Cl*(_{τ}*H*)). Then we have*a*∊̃_{x}*ω*-_{s}*int*(1_{τ}–_{A}*H*) ∈*ω*(_{s}*X, τ, A*) and*a*∊̃_{x}*ω*-_{s}*Cl*(_{τ}*H*). Thus by Theorem 3.16*ω*-_{s}*int*(1_{τ}–_{A}*H*) ∩̃*H*≠ 0. But_{A}*ω*-_{s}*int*(1_{τ}–_{A}*H*) ∩̃*H*⊆̃ (1–_{A}*H*) ∩̃*H*= 0, a contradiction._{A}(b) It is sufficient to show that

$${1}_{A}-{\omega}_{s}-{Cl}_{\tau}(H)\tilde{\subseteq}{\omega}_{s}-{int}_{\tau}({1}_{A}-H).$$

Let

*a*∊̃ (1_{x}–_{A}*ω*-_{s}*Cl*(_{τ}*H*)), then by Theorem 3.16 there is*K*∈*ω*(_{s}*X, τ, A*) such that*a*∊̃_{x}*K*and*K*∩̃*H*= 0. So, we have_{A}*a*∊̃_{x}*K*⊆̃ 1–_{A}*H*and hence*a*∈_{x}*ω*-_{s}*int*(1_{τ}–_{A}*H*).(c) Follows from (a) and (b).

- 4. Relative Topological Spaces and Generated Soft Topological Spaces
At this stage, we believe that the following four questions are natural:

### Question 4.1

Let (

*X, τ, A*) and let*K*∈*ω*(_{s}*X, τ, A*). Is it true that*K*(*a*) ∈*ω*(_{s}*X, τ*) for all_{a}*a*∈*A*?### Question 4.2

Let (

*X, τ, A*) and let*K*∈*SS*(*X, A*) such that*K*(*a*) ∈*ω*(_{s}*X, τ*) for all_{a}*a*∈*A*. Is it true that*K*∈*ω*(_{s}*X, τ, A*)?### Question 4.3

Let (

*X, τ, A*) and let*K*∈*SO*(*X, τ, A*). Is it true that*K*(*a*) ∈*SO*(*X, τ*) for all_{a}*a*∈*A*?### Question 4.4

Let (

*X, τ, A*) and let*K*∈*SS*(*X, A*) such that*K*(*a*) ∈*SO*(*X, τ*) for all_{a}*a*∈*A*. Is it true that*K*∈*SO*(*X, τ, A*)?We will leave Question 4.3 as an open question. However, in the following example we will give negative answers for Questions 4.1, 4.2, and 4.4:

### Example 4.5

Let

*X*= ℝ and*A*= {*a, b*}. Let*F, G, H, K*∈*SS*(*X, A*) defined by$$\begin{array}{l}F(a)={\mathbb{Q}}^{c},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}F\hspace{0.17em}(b)=\left\{1\right\},\\ G(a)=\left\{2\right\},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}G\hspace{0.17em}(b)={\mathbb{Q}}^{c},\\ H(a)={\mathbb{Q}}^{c},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}H\hspace{0.17em}(b)=\mathbb{N},\\ K(a)=\left\{2\right\},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}K\hspace{0.17em}(b)=\left\{1\right\}.\end{array}$$

Let

*τ*= {0_{A}*,*1_{A}*, F, G, F*∪̃*G*}. As clearly that (*X, τ, A*) is soft anti-locally countable, then by Theorem 2.6*ω*(_{s}*X, τ, A*) =*SO*(*X, τ, A*). It is not difficult to check that*H*∈*SO*(*X, τ, A*) =*ω*(_{s}*X, τ, A*) while*H*(*b*) ∉*ω*(_{s}*X, τ*), which gives a negative answer for Questions 4.1. Also it is not difficult to check that_{b}*K*(*a*) ∈*ω*(_{s}*X, τ*) ⊆_{a}*SO*(*X, τ*) and_{a}*K*(*b*) ∈*ω*(_{s}*X, τ*) ⊆_{b}*SO*(*X, τ*) while_{b}*K*∉*ω*(_{s}*X, τ, A*) =*SO*(*X, τ, A*), which gives negative answers for Questions 4.2 and 4.4.If we add the condition ‘(

*X, τ, A*) is soft locally countable’, then Question 4.1 will have a positive answer:### Theorem 4.6

Let (

*X, τ, A*) be a soft locally countable and let*K*∈*ω*(_{s}*X, τ, A*). Then*K*(*a*) ∈*ω*(_{s}*X, τ*) for every_{a}*a*∈*A*.### Proof

Let

*a*∈*A*. As (*X, τ, A*) is soft locally countable and*K*∈*ω*(_{s}*X, τ, A*), then by Theorem 2.8,*K*∈*τ*and so*K*(*a*) ∈*τ*⊆_{a}*ω*(_{s}*X, τ*)._{a}### Lemma 4.7

Let {(

*X*, ℑ) :_{a}*a*∈*A*} be an indexed family of topological spaces and let$\tau =\underset{a\in A}{\oplus}{\Im}_{a}$ . Let*H*∈*SS*(*X, A*), then for every*a*∈*A*,*Cl*_{(}_{τ}_{a}_{)}_{ω}(*H*(*a*)) = (*Cl*_{τ}_{ω}(*H*)) (*a*).### Proof

Let

*a*∈*A*. By Proposition 7 of [4], *Cl*_{(}_{τ}_{a}_{)}_{ω}(*H*(*a*)) ⊆ (*Cl*_{τ}_{ω}(*H*)) (*a*). To see that (*Cl*_{τ}_{ω}(*H*)) (*a*) ⊆*Cl*_{(}_{τ}_{a}_{)}_{ω}(*H*(*a*)), let*x*∈ (*Cl*_{τ}_{ω}(*H*)) (*a*) and let*U*∈ (*τ*)_{a}such that_{ω}*x*∈*U*. By Theorem 7 of [2], ( *τ*)_{a}= (_{ω}*τ*)_{ω}. Then_{a}*a*∈_{U}*τ*. As_{ω}*a*∈_{x}*a*∈_{U}*τ*and_{ω}*a*∈_{x}*Cl*_{τ}_{ω}(*H*), then by Lemma 5 of [2], *H*∩̃*a*≠ 0_{U}. Thus, we must have (_{A}*H*∩̃*a*) (_{U}*a*) =*H*(*a*) ∩*U*≠ ∅︀. It follows that*x*∈*Cl*_{(}_{τ}_{ω}_{)}_{a}(*H*(*a*)).If we add the condition ‘

*τ*is a generated soft topology’, then Questions 4.1 and 4.2 will have positive answers:### Theorem 4.8

Let {(

*X*, ℑ) :_{a}*a*∈*A*} be an indexed family of topological spaces and let$\tau =\underset{a\in A}{\oplus}{\Im}_{a}$ . Let*H*∈*SS*(*X, A*). Then*H*∈*ω*(_{s}*X, τ, A*) if and only if*H*(*a*) ∈*ω*(_{s}*X, τ*) for every_{a}*a*∈*A*.### Proof

### Necessity

Suppose that

*H*∈*ω*(_{s}*X, τ, A*) and let*a*∈*A*. Choose*F*∈*τ*such that*F*⊆̃*H*⊆̃*Cl*_{τ}_{ω}(*F*). So,*F*(*a*) ⊆*H*(*a*) ⊆ (*Cl*_{τ}_{ω}(*F*)) (*a*). As*F*(*a*) ∈*τ*and by Lemma 4.7 we have (_{a}*Cl*_{τ}_{ω}(*F*)) (*a*) =*Cl*_{(}_{τ}_{a}_{)}_{ω}(*F*(*a*)), then*H*(*a*) ∈*ω*(_{s}*X, τ*)._{a}### Sufficiency

Suppose that

*H*(*a*) ∈*ω*(_{s}*X, τ*) for every_{a}*a*∈*A*. Then for every*a*∈*A*, there exists*U*∈_{a}*τ*= ℑ_{a}such that_{a}*U*⊆_{a}*H*(*a*) ⊆*Cl*_{(}_{τ}_{a}_{)}_{ω}(*U*). Let_{a}*K*∈*SS*(*X, A*) with*K*(*a*) =*U*∈ ℑ_{a}for every_{a}*a*∈*A*. Then$K\in \underset{a\in A}{\oplus}{\Im}_{a}=\tau $ and by Lemma 4.7, (

*Cl*_{τ}_{ω}(*K*)) (*a*) =*Cl*_{(}_{τ}_{a)ω}(*K*(*a*)) =*Cl*_{(}_{τ}_{a)ω}(*U*) for all_{a}*a*∈*A*. As for every*a*∈*A*,*K*(*a*) =*U*⊆_{a}*H*(*a*) ⊆*Cl*_{(}_{τ}_{a)ω}(*U*) = (_{a}*Cl*_{τ}_{ω}(*K*)) (*a*), then*K*⊆̃*H*⊆̃*Cl*_{τ}_{ω}(*K*). It follows that*H*∈*ω*(_{s}*X, τ, A*).### Lemma 4.9

Let {(

*X*, ℑ) :_{a}*a*∈*A*} be an indexed family of topological spaces and let$\tau =\underset{a\in A}{\oplus}{\Im}_{a}$ . Let*H*∈*SS*(*X, A*), then for every*a*∈*A*,*Cl*_{τ}_{a}(*H*(*a*)) = (*Cl*(_{τ}*H*)) (*a*).### Proof

Let

*a*∈*A*. By Proposition 7 of [4], *Cl*_{τ}_{a}(*H*(*a*)) ⊆ (*Cl*(_{τ}*H*)) (*a*). To see that (*Cl*(_{τ}*H*)) (*a*) ⊆*Cl*_{τ}_{a}(*H*(*a*)), let*x*∈ (*Cl*(_{τ}*H*)) (*a*) and let*U*∈*τ*= ℑ_{a}such that_{a}*x*∈*U*. Then we have*a*∊̃_{x}*a*∈_{U}*τ*and*a*∊̃_{x}*Cl*(_{τ}*H*), and by Lemma 5 of [2], *H*∩̃*a*≠ 0_{U}. Thus, we must have (_{A}*H*∩̃*a*) (_{U}*a*) =*H*(*a*) ∩*U*≠ ∅︀. It follows that*x*∈*Cl*_{τ}_{a}(*H*(*a*)).If we add the condition ‘

*τ*is a generated soft topology’, then Questions 4.3 and 4.4 will have positive answers:### Theorem 4.10

Let {(

*X*, ℑ) :_{a}*a*∈*A*} be an indexed family of topological spaces and let$\tau =\underset{a\in A}{\oplus}{\Im}_{a}$ . Let*H*∈*SS*(*X, A*). Then*H*∈*SO*(*X, τ, A*) if and only if*H*(*a*) ∈*SO*(*X, τ*) for every_{a}*a*∈*A*.### Proof

### Necessity

Suppose that

*H*∈*SO*(*X, τ, A*). Then there exists*F*∈*τ*such that*F*⊆̃*H*⊆̃*Cl*(_{τ}*F*). So,*F*(*a*) ⊆*H*(*a*) ⊆ (*Cl*(_{τ}*F*)) (*a*). As*F*(*a*) ∈*τ*and by Lemma 4.9 we have (_{a}*Cl*(_{τ}*F*)) (*a*) =*Cl*_{τ}_{a}(*F*(*a*)), then*H*(*a*) ∈*SO*(*X, τ*)._{a}### Sufficiency

Suppose that

*H*(*a*) ∈*SO*(*X, τ*) for every_{a}*a*∈*A*. Then for every*a*∈*A*, there exists*U*∈_{a}*τ*= ℑ_{a}such that_{a}*U*⊆_{a}*H*(*a*) ⊆*Cl*_{τ}_{a}(*U*). Let_{a}*K*∈*SS*(*X, A*) with*K*(*a*) =*U*∈ ℑ_{a}for every_{a}*a*∈*A*. Then$K\in \underset{a\in A}{\oplus}{\Im}_{a}=\tau $ and by Lemma 4.9, (*Cl*(_{τ}*K*)) (*a*) =*Cl*_{τ}_{a}(*K*(*a*)) =*Cl*_{τ}_{a}(*U*) for all_{a}*a*∈*A*. As for every*a*∈*A*,*K*(*a*) =*U*⊆_{a}*H*(*a*) ⊆*Cl*_{τ}_{a}(*U*) = (_{a}*Cl*(_{τ}*K*)) (*a*), then*K*⊆̃*H*⊆̃*Cl*(_{τ}*K*). It follows that*H*∈*SO*(*X, τ, A*).

- 5. Conclusion

We introduced soft*ω*-open sets as a new class of soft sets. We then proved that the class of soft_{s}*ω*-open sets lies strictly between the classes of soft open sets and soft semi-open sets. Furthermore, we studied two new soft operators, soft_{s}*ω*-closure and soft_{s}*ω*-interior, as well as relationships regarding relative topological spaces and generated soft topological. Finally, we end by raising an open question related to soft_{s}*ω*-open. We believe the proposed concept can open up doors to research areas such as soft_{s}*ω*-mappings and soft_{s}*ω*-compactness._{s}

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

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

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- Biography
Samer Al Ghour received the Ph.D. in Mathematics from University of Jordan, Jordan in 1999. Currently, he is currently a professor at the Department of Mathematics and Statistics, Jordan University of Science and Technology, Jordan. His research interests is include general topology, fuzzy topology, and soft set theory.E-mail: algore@just.edu.jo