International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(2): 181-191

**Published online** June 25, 2023

https://doi.org/10.5391/IJFIS.2023.23.2.181

© The Korean Institute of Intelligent Systems

Samer Al Ghour

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

**Correspondence to : **

Samer Al Ghour (algore@just.edu.jo)

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 study, we obtained several results regarding the soft *wb*-open sets. For example, we demonstrate that they form a soft supra topology that contains classes of soft *w*-open sets and soft *b*-open sets. Additionally, using soft *wb*-open sets, we define and investigate the soft *wb*-closure and soft *wb*-interior as two new operators. Furthermore, we introduce and investigate soft *b*-antilocal countability as a novel soft-topological property. In addition, we introduce quasi-soft *b*-openness and weakly quasi-soft *b*-openness as two new classes of soft functions. Finally, we investigate the relationships between the new concepts and their analogs in a general topology.

**Keywords**: Soft *w*-open sets, Soft *b*-open sets, Soft *b*-Lindelof generates a soft topology

This study follows the concepts and terminology in [1, 2]. In this study, STS and TS denote the soft topological space and topological space, respectively. As a general mathematical tool for addressing uncertainty, Molodtsov [3] introduced the notion of soft sets in 1999. Let _{E}_{E}_{E}_{E}_{E}^{c}

As a weaker form of the semi-open and preopen sets, the notion of

The remainder of this paper is organized as follows.

In Section 2, we introduce several properties of soft

In Section 3, we establish the characterization and preservation theorems for soft

Let (_{τ}_{τ}_{μ}_{μ}

The following definitions and results were used:

Let (

(a) _{μ}_{μ}_{μ}_{μ}

(b)

The families of all

Let (

(a) an

(b) an

The families of all

Let (

(a) soft

(b) soft _{E}

The families of all soft

Let (

(a) The soft _{τ}

(b) The soft _{τ}

The function

(a) [34] quasi

(b) [34] weakly quasi

(c) [40]

(d) [34]

Let (

(a)

(b) (

A soft function _{pu}

Here, we introduce several properties of soft

A soft set _{y}_{y}

The families of all soft

For any STS (_{ω}

Let _{ω}_{y}_{y}

If (

From Corollary 5 of [2], _{ω}

For any STS (

Let _{y}_{y}

For any STS (

The following example shows that inclusion in Corollary 2.5 cannot be replaced by equality. In general,

Let

Therefore, _{{3}}_{{3}} ∈

The following example shows that inclusion in Theorem 2.2 cannot be replaced by equality. In general,

Let _{ℚ}. As _{τ}_{Cl}_{μ(ℚ)} = _{ℝ} = 1_{E}_{τ}_{τ}_{τ}_{τ}_{ω}

Let (_{y}_{y}

_{y}_{y}

Let _{y}_{y}

Let (

Suppose that _{E}_{E}_{E}_{y}_{E}_{y}_{E}_{E}_{E}_{E}_{E}_{E}

Let (_{ω}

Let _{y}_{y}_{y}_{y}_{ω}_{y}_{y}

then (

Let (

For a STS (

Let

Then _{ℚ}, _{[0,1)} ∈ _{ℚ} ∩̃ _{[0,1)} = _{ℚ∩[0,1)} ∈_{0} ∊̃_{ℚ∩[0,1)} ∈ _{ℚ∩[0,1)} ∈ _{ℚ} ∩̃_{[0,1)}

Let (_{α}_{α}_{∈Δ}_{α}

Let _{y}_{α}_{∈Δ}_{α}_{y}_{β}_{y}_{β}

As _{β}_{α}_{∈Δ}_{α}_{α}_{∈Δ}_{α}

Let (_{α}_{α}_{∈Δ}_{α}

Let {_{α}_{E}_{α}_{α}_{∈Δ}(1_{E}_{α}_{E}_{α}_{∈Δ}_{α}_{α}_{∈Δ}_{α}

Let {(_{e}_{e}_{∈}_{E}_{e}_{e}

_{e}_{∈}_{E}_{e}_{e}_{∈}_{E}_{e}_{e}_{∈}_{E}_{e}_{τ}_{τ}_{τ}_{τ}

According to Lemma 4.9 of [19],

and

Therefore,

Hence, _{e}

Let _{e}

for all

and

Hence,

Therefore, _{e}_{∈}_{E}_{e}

Let (

For each _{e}_{e}_{∈}_{E}_{e}

Let {(_{e}_{e}_{∈}_{E}_{e}_{e}

Let _{e}_{∈}_{E}_{e}_{y}_{e}_{∈}_{E}_{e}_{y}_{e}_{e}

Suppose _{e}_{y}_{e}_{e}_{y}_{V}_{V}_{e}_{∈}_{E}_{e}_{V}_{e}_{∈}_{E}_{e}

Let (

For each _{e}_{e}_{∈}_{E}_{e}

Let (

(a) The soft _{τ}

(b) The soft _{τ}

Let (

(a) _{τ}

(b) _{τ}

Obvious.

Let (

(a) _{τ}

(b) _{τ}

(c) _{y}_{τ}_{y}_{E}

(a) and (b) are significant.

(c) Suppose that _{y}_{τ}_{y}_{E}_{E}_{τ}_{E}_{y}_{τ}_{y}_{E}_{y}_{y}_{E}_{y}_{E}_{τ}_{τ}_{E}_{τ}_{E}_{τ}_{E}_{E}_{τ}_{E}_{E}

Let (

(a) _{τ}_{E}_{E}_{τ}

(b) _{τ}_{E}_{E}_{τ}

(a) As 1_{E}_{E}_{τ}_{τ}_{E}_{E}_{τ}_{E}_{τ}_{τ}_{E}_{y}_{E}_{τ}_{τ}_{E}_{y}_{E}_{τ}_{E}_{y}_{E}_{E}_{y}_{y}_{τ}_{y}_{E}_{τ}

(b) By (a), _{τ}_{E}_{E}_{τ}_{E}_{E}_{E}_{τ}

Let (_{τ}_{τ}

Following Corollary 2.3.

Let (

(a) _{τ}_{τ}

(b) _{τ}_{τ}

Following these definitions and Corollary 2.5.

The following examples show that each inclusion in Theorem 2.24 cannot be replaced by an equality.

Let _{{2}}. As _{τ}_{τ}_{τ}_{τ}_{E}_{τ}_{E}_{τ}_{E}_{E}_{τ}_{E}_{τ}_{τ}_{τ}_{E}_{E}_{τ}_{E}

STS (_{E}

Every soft

Follows from the definitions and the fact that

The following example shows that the converse of Theorem 2.27 need not be true in general:

Let (_{ℚ}. It is proved in Example 2.7 that _{E}

Let {(_{e}_{e}_{∈}_{E}_{e}_{e}

Suppose that (_{e}_{∈}_{E}_{e}_{e}_{U}_{e}_{U}_{e}_{∈}_{E}_{e}_{U}_{U}_{e}_{∈}_{E}_{e}_{E}_{e}_{∈}_{E}_{e}_{U}_{U}_{U}_{e}

Suppose that (_{e}_{e}_{∈}_{E}_{e}_{E}_{e}_{e}_{∈}_{E}_{e}

Let (

For each _{e}_{e}_{∈}_{E}_{e}

Let (_{τ}_{τ}

By Theorem 2.24 (a), _{τ}_{τ}_{τ}_{τ}_{y}_{τ}_{y}_{y}_{y}_{τ}_{y}_{E}_{E}_{y}_{τ}

Let (_{τ}_{τ}

By Theorem 2.24 (a), _{τ}_{τ}_{τ}_{τ}_{y}_{τ}_{y}_{y}_{y}_{τ}_{y}_{E}_{E}_{y}_{τ}

Let (^{c}_{τ}_{τ}

As ^{c}_{E}_{τ}_{E}_{τ}_{E}_{E}_{τ}_{E}_{E}_{τ}_{E}_{E}_{τ}_{E}_{τ}_{E}_{τ}_{E}_{τ}_{E}_{E}_{τ}_{τ}_{τ}

A soft function _{pu}

(a) quasi soft _{pu}

(b) weakly quasi soft _{pu}

Every quasi soft

Straightforward.

The following is an example of a quasi soft

Let (

Subsequently, _{pu}

The following example shows that the converse of Theorem 2.34 need not be true in general:

Let (

Let _{pu}_{pu}_{pu}_{pu}

Let _{pu}

Suppose _{pu}_{U}_{pu}_{U}_{p}_{(}_{U}_{)} ∈ _{p}_{(}_{U}_{)}) (

Suppose that _{pu}_{pu}_{pu}

Quasi soft

Let _{pu}_{pu}

Let _{pu}

Suppose _{pu}_{U}_{pu}_{U}_{p}_{(}_{U}_{)} ∈ _{p}_{(}_{U}_{)})(

Suppose that _{pu}_{pu}_{pu}

Let _{pu}_{pu}_{ω}

Let _{z}_{pu}_{y}_{pu}_{y}_{z}_{y}_{pu}_{pu}_{z}_{pu}_{y}_{pu}_{pu}_{pu}_{pu}_{pu}_{ω}

Here, we establish the characterization and preservation theorems for soft

Let (

(a) _{F}_{∈}_{β}

(b)

(c)

(d) (_{E}

If (

Let _{K}_{∈}_{β}

For any STS (

(a) (

(b) Every soft cover of 1_{E}

(a) ⇒ (b): Let _{E}_{y}_{e}_{y}∈ _{y}_{e}_{y}. As _{y}_{e}_{y}∈ _{y}_{e}_{y}and _{e}_{y}– _{e}_{y}∈ _{e}_{y} : _{y}_{E}_{e}_{y} : _{y}_{E}

As _{e}_{y}– _{e}_{y}∈ _{y}_{e}_{y∈}_{S}_{e}_{y}– _{e}_{y} ) ∈ _{1} of _{e}_{y∈}_{S}_{e}_{y}– _{e}_{y} )⊆̃ ∪̃_{M}_{∈}_{β}_{1}_{2} = {_{e}_{y} : _{y}_{1}. Then _{2} is a countable subcover of

(b) ⇒ (a): Follows from Corollary 2.5.

Let {(_{e}_{e}_{∈}_{E}_{e}_{e}

Let (_{e}_{∈}_{E}_{e}_{Y}_{E}_{Y}_{1}}, where _{1} is a countable subset of _{1} and hence _{e}_{e}_{H}_{∈}_{ℋ}_{H}_{H}_{Y}_{E}_{e}_{∈}_{E}_{e}_{1} of _{H}_{1}}. Thus, _{1} is a countable subcover of _{e}

A soft function _{pu}

Let _{pu}

Suppose _{pu}_{W}_{pu}

Suppose that

Let _{pu}

Suppose _{pu}_{W}_{pu}

Suppose that

Every soft

Following the definitions in Corollary 2.5.

Remark: In general, the opposite of Theorem 3.8 is not true.

Let ^{−1}({1}) = {3} _{pu}

Let _{pu}

Let _{D}_{pu}_{E}_{1} is a countable subset of _{1} is a countable soft subcover of

Let (

Let _{E}_{y}_{E}_{e}_{y}∈ _{y}_{e}_{y} and _{e}_{y}– (1_{E}_{e}_{y}∩̃ _{e}_{y} : _{y}_{E}_{E}_{1} ⊆ _{y}_{y}_{E}_{1} ∪ {_{e}_{y} : _{y}_{E}_{e}_{y∊̃}_{M}_{e}_{y}∩̃ _{z}_{t}_{z}∈ _{z}_{t}_{z}. Hence, _{1} ∪ {_{t}_{z} : _{z}

A soft function _{pu}_{pu}

Let _{pu}_{z}

Let _{E}_{z}_{d}_{z}⊆ _{d}_{z} is a soft cover of _{z}_{d}_{z} = ∪̃_{H}_{∈}_{β}_{dz}_{d}_{z} = 1_{Z}_{pu}_{E}_{d}_{z} ). Then for each _{z}_{z}_{d}_{z} and _{pu}_{d}_{z}∈ _{z}_{d}_{z}∈ _{z}_{d}_{z} and _{d}_{z}–_{d}_{z}∈ _{z}_{d}_{z}⊆̃ (_{d}_{z}– _{d}_{z} )∪̃_{d}_{z} ; therefore,

As _{d}_{z}– _{d}_{z}∈

and hence

As {_{d}_{z} : _{z}_{D}_{d}_{z∈}_{M}_{d}_{z} = 1_{D}

It follows that (

The growth of the topology was supported by a continuous supply of TS classes, examples, properties, and relationships. Therefore, expanding the structure of STSs in the same way is important.

This study aims to examine the behaviors of soft

The following topics can be considered in future studies: 1) introducing soft topological concepts using soft

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

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International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(2): 181-191

**Published online** June 25, 2023 https://doi.org/10.5391/IJFIS.2023.23.2.181

Copyright © The Korean Institute of Intelligent Systems.

Samer Al Ghour

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

**Correspondence to:**Samer Al Ghour (algore@just.edu.jo)

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 study, we obtained several results regarding the soft *wb*-open sets. For example, we demonstrate that they form a soft supra topology that contains classes of soft *w*-open sets and soft *b*-open sets. Additionally, using soft *wb*-open sets, we define and investigate the soft *wb*-closure and soft *wb*-interior as two new operators. Furthermore, we introduce and investigate soft *b*-antilocal countability as a novel soft-topological property. In addition, we introduce quasi-soft *b*-openness and weakly quasi-soft *b*-openness as two new classes of soft functions. Finally, we investigate the relationships between the new concepts and their analogs in a general topology.

**Keywords**: Soft *w*-open sets, Soft *b*-open sets, Soft *b*-Lindelof generates a soft topology

This study follows the concepts and terminology in [1, 2]. In this study, STS and TS denote the soft topological space and topological space, respectively. As a general mathematical tool for addressing uncertainty, Molodtsov [3] introduced the notion of soft sets in 1999. Let _{E}_{E}_{E}_{E}_{E}^{c}

As a weaker form of the semi-open and preopen sets, the notion of

The remainder of this paper is organized as follows.

In Section 2, we introduce several properties of soft

In Section 3, we establish the characterization and preservation theorems for soft

Let (_{τ}_{τ}_{μ}_{μ}

The following definitions and results were used:

Let (

(a) _{μ}_{μ}_{μ}_{μ}

(b)

The families of all

Let (

(a) an

(b) an

The families of all

Let (

(a) soft

(b) soft _{E}

The families of all soft

Let (

(a) The soft _{τ}

(b) The soft _{τ}

The function

(a) [34] quasi

(b) [34] weakly quasi

(c) [40]

(d) [34]

Let (

(a)

(b) (

A soft function _{pu}

Here, we introduce several properties of soft

A soft set _{y}_{y}

The families of all soft

For any STS (_{ω}

Let _{ω}_{y}_{y}

If (

From Corollary 5 of [2], _{ω}

For any STS (

Let _{y}_{y}

For any STS (

The following example shows that inclusion in Corollary 2.5 cannot be replaced by equality. In general,

Let

Therefore, _{{3}}_{{3}} ∈

The following example shows that inclusion in Theorem 2.2 cannot be replaced by equality. In general,

Let _{ℚ}. As _{τ}_{Cl}_{μ(ℚ)} = _{ℝ} = 1_{E}_{τ}_{τ}_{τ}_{τ}_{ω}

Let (_{y}_{y}

_{y}_{y}

Let _{y}_{y}

Let (

Suppose that _{E}_{E}_{E}_{y}_{E}_{y}_{E}_{E}_{E}_{E}_{E}_{E}

Let (_{ω}

Let _{y}_{y}_{y}_{y}_{ω}_{y}_{y}

then (

Let (

For a STS (

Let

Then _{ℚ}, _{[0,1)} ∈ _{ℚ} ∩̃ _{[0,1)} = _{ℚ∩[0,1)} ∈_{0} ∊̃_{ℚ∩[0,1)} ∈ _{ℚ∩[0,1)} ∈ _{ℚ} ∩̃_{[0,1)}

Let (_{α}_{α}_{∈Δ}_{α}

Let _{y}_{α}_{∈Δ}_{α}_{y}_{β}_{y}_{β}

As _{β}_{α}_{∈Δ}_{α}_{α}_{∈Δ}_{α}

Let (_{α}_{α}_{∈Δ}_{α}

Let {_{α}_{E}_{α}_{α}_{∈Δ}(1_{E}_{α}_{E}_{α}_{∈Δ}_{α}_{α}_{∈Δ}_{α}

Let {(_{e}_{e}_{∈}_{E}_{e}_{e}

_{e}_{∈}_{E}_{e}_{e}_{∈}_{E}_{e}_{e}_{∈}_{E}_{e}_{τ}_{τ}_{τ}_{τ}

According to Lemma 4.9 of [19],

and

Therefore,

Hence, _{e}

Let _{e}

for all

and

Hence,

Therefore, _{e}_{∈}_{E}_{e}

Let (

For each _{e}_{e}_{∈}_{E}_{e}

Let {(_{e}_{e}_{∈}_{E}_{e}_{e}

Let _{e}_{∈}_{E}_{e}_{y}_{e}_{∈}_{E}_{e}_{y}_{e}_{e}

Suppose _{e}_{y}_{e}_{e}_{y}_{V}_{V}_{e}_{∈}_{E}_{e}_{V}_{e}_{∈}_{E}_{e}

Let (

For each _{e}_{e}_{∈}_{E}_{e}

Let (

(a) The soft _{τ}

(b) The soft _{τ}

Let (

(a) _{τ}

(b) _{τ}

Obvious.

Let (

(a) _{τ}

(b) _{τ}

(c) _{y}_{τ}_{y}_{E}

(a) and (b) are significant.

(c) Suppose that _{y}_{τ}_{y}_{E}_{E}_{τ}_{E}_{y}_{τ}_{y}_{E}_{y}_{y}_{E}_{y}_{E}_{τ}_{τ}_{E}_{τ}_{E}_{τ}_{E}_{E}_{τ}_{E}_{E}

Let (

(a) _{τ}_{E}_{E}_{τ}

(b) _{τ}_{E}_{E}_{τ}

(a) As 1_{E}_{E}_{τ}_{τ}_{E}_{E}_{τ}_{E}_{τ}_{τ}_{E}_{y}_{E}_{τ}_{τ}_{E}_{y}_{E}_{τ}_{E}_{y}_{E}_{E}_{y}_{y}_{τ}_{y}_{E}_{τ}

(b) By (a), _{τ}_{E}_{E}_{τ}_{E}_{E}_{E}_{τ}

Let (_{τ}_{τ}

Following Corollary 2.3.

Let (

(a) _{τ}_{τ}

(b) _{τ}_{τ}

Following these definitions and Corollary 2.5.

The following examples show that each inclusion in Theorem 2.24 cannot be replaced by an equality.

Let _{{2}}. As _{τ}_{τ}_{τ}_{τ}_{E}_{τ}_{E}_{τ}_{E}_{E}_{τ}_{E}_{τ}_{τ}_{τ}_{E}_{E}_{τ}_{E}

STS (_{E}

Every soft

Follows from the definitions and the fact that

The following example shows that the converse of Theorem 2.27 need not be true in general:

Let (_{ℚ}. It is proved in Example 2.7 that _{E}

Let {(_{e}_{e}_{∈}_{E}_{e}_{e}

Suppose that (_{e}_{∈}_{E}_{e}_{e}_{U}_{e}_{U}_{e}_{∈}_{E}_{e}_{U}_{U}_{e}_{∈}_{E}_{e}_{E}_{e}_{∈}_{E}_{e}_{U}_{U}_{U}_{e}

Suppose that (_{e}_{e}_{∈}_{E}_{e}_{E}_{e}_{e}_{∈}_{E}_{e}

Let (

For each _{e}_{e}_{∈}_{E}_{e}

Let (_{τ}_{τ}

By Theorem 2.24 (a), _{τ}_{τ}_{τ}_{τ}_{y}_{τ}_{y}_{y}_{y}_{τ}_{y}_{E}_{E}_{y}_{τ}

_{τ}_{τ}

_{τ}_{τ}_{τ}_{τ}_{y}_{τ}_{y}_{y}_{y}_{τ}_{y}_{E}_{E}_{y}_{τ}

Let (^{c}_{τ}_{τ}

As ^{c}_{E}_{τ}_{E}_{τ}_{E}_{E}_{τ}_{E}_{E}_{τ}_{E}_{E}_{τ}_{E}_{τ}_{E}_{τ}_{E}_{τ}_{E}_{E}_{τ}_{τ}_{τ}

A soft function _{pu}

(a) quasi soft _{pu}

(b) weakly quasi soft _{pu}

Every quasi soft

Straightforward.

The following is an example of a quasi soft

Let (

Subsequently, _{pu}

The following example shows that the converse of Theorem 2.34 need not be true in general:

Let (

Let _{pu}_{pu}_{pu}_{pu}

Let _{pu}

Suppose _{pu}_{U}_{pu}_{U}_{p}_{(}_{U}_{)} ∈ _{p}_{(}_{U}_{)}) (

Suppose that _{pu}_{pu}_{pu}

Quasi soft

Let _{pu}_{pu}

Let _{pu}

Suppose _{pu}_{U}_{pu}_{U}_{p}_{(}_{U}_{)} ∈ _{p}_{(}_{U}_{)})(

Suppose that _{pu}_{pu}_{pu}

Let _{pu}_{pu}_{ω}

Let _{z}_{pu}_{y}_{pu}_{y}_{z}_{y}_{pu}_{pu}_{z}_{pu}_{y}_{pu}_{pu}_{pu}_{pu}_{pu}_{ω}

Here, we establish the characterization and preservation theorems for soft

Let (

(a) _{F}_{∈}_{β}

(b)

(c)

(d) (_{E}

If (

Let _{K}_{∈}_{β}

For any STS (

(a) (

(b) Every soft cover of 1_{E}

(a) ⇒ (b): Let _{E}_{y}_{e}_{y}∈ _{y}_{e}_{y}. As _{y}_{e}_{y}∈ _{y}_{e}_{y}and _{e}_{y}– _{e}_{y}∈ _{e}_{y} : _{y}_{E}_{e}_{y} : _{y}_{E}

As _{e}_{y}– _{e}_{y}∈ _{y}_{e}_{y∈}_{S}_{e}_{y}– _{e}_{y} ) ∈ _{1} of _{e}_{y∈}_{S}_{e}_{y}– _{e}_{y} )⊆̃ ∪̃_{M}_{∈}_{β}_{1}_{2} = {_{e}_{y} : _{y}_{1}. Then _{2} is a countable subcover of

(b) ⇒ (a): Follows from Corollary 2.5.

Let {(_{e}_{e}_{∈}_{E}_{e}_{e}

Let (_{e}_{∈}_{E}_{e}_{Y}_{E}_{Y}_{1}}, where _{1} is a countable subset of _{1} and hence _{e}_{e}_{H}_{∈}_{ℋ}_{H}_{H}_{Y}_{E}_{e}_{∈}_{E}_{e}_{1} of _{H}_{1}}. Thus, _{1} is a countable subcover of _{e}

A soft function _{pu}

Let _{pu}

Suppose _{pu}_{W}_{pu}

Suppose that

Let _{pu}

Suppose _{pu}_{W}_{pu}

Suppose that

Every soft

Following the definitions in Corollary 2.5.

Remark: In general, the opposite of Theorem 3.8 is not true.

Let ^{−1}({1}) = {3} _{pu}

Let _{pu}

Let _{D}_{pu}_{E}_{1} is a countable subset of _{1} is a countable soft subcover of

Let (

Let _{E}_{y}_{E}_{e}_{y}∈ _{y}_{e}_{y} and _{e}_{y}– (1_{E}_{e}_{y}∩̃ _{e}_{y} : _{y}_{E}_{E}_{1} ⊆ _{y}_{y}_{E}_{1} ∪ {_{e}_{y} : _{y}_{E}_{e}_{y∊̃}_{M}_{e}_{y}∩̃ _{z}_{t}_{z}∈ _{z}_{t}_{z}. Hence, _{1} ∪ {_{t}_{z} : _{z}

A soft function _{pu}_{pu}

Let _{pu}_{z}

Let _{E}_{z}_{d}_{z}⊆ _{d}_{z} is a soft cover of _{z}_{d}_{z} = ∪̃_{H}_{∈}_{β}_{dz}_{d}_{z} = 1_{Z}_{pu}_{E}_{d}_{z} ). Then for each _{z}_{z}_{d}_{z} and _{pu}_{d}_{z}∈ _{z}_{d}_{z}∈ _{z}_{d}_{z} and _{d}_{z}–_{d}_{z}∈ _{z}_{d}_{z}⊆̃ (_{d}_{z}– _{d}_{z} )∪̃_{d}_{z} ; therefore,

As _{d}_{z}– _{d}_{z}∈

and hence

As {_{d}_{z} : _{z}_{D}_{d}_{z∈}_{M}_{d}_{z} = 1_{D}

It follows that (

The growth of the topology was supported by a continuous supply of TS classes, examples, properties, and relationships. Therefore, expanding the structure of STSs in the same way is important.

This study aims to examine the behaviors of soft

The following topics can be considered in future studies: 1) introducing soft topological concepts using soft

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