International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(1): 89-99

**Published online** March 25, 2022

https://doi.org/10.5391/IJFIS.2022.22.1.89

© 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 (Samer Al Ghour)

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.

The soft *θ*_{ω}-closure operator is defined as a new soft operator that lies strictly between the usual soft closure and the soft *θ*-closure. Sufficient conditions are provided for equivalence between the soft *θ*_{ω}-closure and usual soft closure operators, and between the soft *θ*_{ω}-closure and soft *θ*-closure operators. Via the soft *θ*_{ω}-closure operator, the soft *θ*_{ω}-open sets are defined as a new class of soft sets that lies strictly between the class of soft open sets and the class of soft *θ*-open sets. It is proven that the class of soft *θ*_{ω}-open sets form a new soft topology. The soft ω-regularity is characterized via both the soft *θ*_{ω}-closure operator and soft *θ*_{ω}-open sets. The soft product theorem and several soft mapping theorems are introduced. The correspondence between the soft topology of the soft *θ*_{ω}-open sets of soft topological space and their generated topological spaces, and vice versa, are studied. In addition to these, soft *θ*_{ω}-continuity as a strong form of soft *θ*-continuity is introduced and investigated.

**Keywords**: Soft *θ*-closure, Soft *θ*-open, Soft ω-regular, Soft product, Soft *θ*-continuity, Soft generated soft topological space.

This paper follows the concepts and terminology that appear in [1–3]. In this paper, TS and STS denote the topological space and soft topological space, respectively. The concept of soft sets, which was introduced by Molodtsov [4] in 1999, is a general mathematical tool for dealing with uncertainty. Let _{A}_{A}_{A}_{A}_{A}_{ω}_{ω}_{ω}_{ω}_{ω}_{ω}_{ω}_{ω}_{ω}_{ω}_{ω}_{ω}

The following definitions and results are used throughout this work:

Let

(a) [1] _{Y}

(b) [1] _{Y}

(c) [46] _{x}

The set of all soft points in

Let _{x}_{x}_{x}_{x}_{x}

Let (_{a}

Let (

defines a soft topology on

Let _{a}

Then,

Let

for each (

Let (

Recall that a soft set

An STS (

(a) [50] soft locally indiscrete if every soft open set is soft closed;

(b) [2] soft locally countable if for each _{x}_{x}

(c) [2] soft anti-locally countable if for every _{A}

(d) [6] soft

(e) [6] soft _{x}_{A}_{ω}_{x}_{A}

In this section, we introduce the _{ω}_{ω}_{ω}_{ω}_{ω}_{ω}_{ω}_{ω}_{ω}

Let (

(a) A soft point _{x}_{x}_{θ}_{τ}_{A}_{x}

(b) _{θ}

(c) _{A}

(d) The family of all soft _{θ}

Let (

(a) (_{θ}

(b) _{θ}_{θ}

The following is the main definition of this work.

Let (

(a) A soft point _{x}_{ω}_{x}_{θ}_{ω} (_{τ}_{ω} (_{A}_{x}

(b) _{ω}_{θ}_{ω} (

(c) _{ω}_{A}_{ω}

(d) The family of all soft _{ω}_{θ}_{ω}.

Let (

(a) _{τ}_{θ}_{ω} (_{θ}

(b) If _{ω}

(c) If _{ω}

(a) To demonstrate that _{τ}_{θ}_{ω} (_{x}_{τ}_{x}_{x}_{τ}_{A}_{τ}_{ω} (_{τ}_{ω} (_{A}_{x}_{θ}_{ω} (_{θ}_{ω} (_{θ}_{x}_{θ}_{ω} (_{x}_{x}_{θ}_{ω} (_{τ}_{ω} (_{A}_{τ}_{ω} (_{τ}_{τ}_{A}_{x}_{θ}

(b) Suppose that _{θ}_{θ}_{ω} (_{ω}

(c) Suppose that _{ω}_{θ}_{ω} (_{τ}

Let (

(a) _{τ}_{θ}_{ω} (F).

(b) If _{ω}

(a) From Theorem 2.4 (a), _{τ}_{θ}_{ω} (_{θ}_{ω} (_{τ}_{x}_{θ}_{ω} (_{x}_{τ}_{ω} (_{A}_{τ}_{ω} (_{A}_{x}_{τ}

(b) Suppose that _{τ}_{θ}_{ω} (_{ω}

Let (

(a) _{τ}_{θ}_{ω} (

(b) If _{ω}

The proof follows from Theorem 2.5 and Theorem 7 of [6].

Let (

(a) _{τ}_{θ}_{ω} (

(b) If _{ω}

The proof follows from Theorem 2.5 and Theorem 6 of [6].

Let (

(a) _{θ}_{θ}_{ω} (

(b) If _{ω}

(a) Using Theorem 2.4(a), _{θ}_{ω} (_{θ}_{θ}_{θ}_{ω} (_{x}_{θ}_{x}_{τ}_{A}_{τ}_{ω} (_{τ}_{τ}_{ω} (_{A}_{x}_{θ}_{ω} (

(b) Suppose that _{ω}_{θ}_{ω} (_{θ}

For any STS (_{θ}_{θ}_{ω} ⊆

Let _{θ}_{A}_{A}_{ω}_{θ}_{ω}. Therefore, _{θ}_{θ}_{ω}. To demonstrate that _{θ}_{ω} ⊆ _{θ}_{ω}; then, 1_{A}_{ω}_{A}

Let (_{θ}_{τ}

Let _{τ}_{θ}_{θ}_{τ}_{x}_{θ}_{x}_{τ}_{A}_{y}_{τ}_{A}_{τ}_{θ}

Let (

(a) If _{A}_{θ}_{ω} (_{θ}_{ω} (

(b) For each _{θ}_{ω} (_{θ}_{ω} (_{θ}_{ω} (

(c) For each _{θ}_{ω} (

(d) For each _{ω}_{θ}_{ω} (_{τ}

(e) For each _{θ}_{θ}_{ω} (_{τ}

(a) Let _{x}_{θ}_{ω} (_{x}_{x}_{θ}_{ω} (_{τ}_{ω} (_{A}_{τ}_{ω} (_{A}_{x}_{θ}_{ω} (

(b) As _{θ}_{ω} (_{θ}_{ω} (_{θ}_{ω} (_{x}_{A}_{θ}_{ω} (_{θ}_{ω} (_{x}_{τ}_{ω} (_{A}_{τ}_{ω} (_{A}_{x}

Hence, _{x}_{A}_{θ}_{ω} (

(c) Let _{A}_{θ}_{ω} (_{x}_{A}_{θ}_{ω} (_{x}_{τ}_{ω} (_{A}_{θ}_{ω} (_{A}_{A}_{θ}_{ω} (

(d) Let _{ω}_{τ}_{θ}_{ω} (_{θ}_{ω} (_{τ}_{x}_{θ}_{ω} (_{A}_{τ}_{x}_{θ}_{ω}(_{x}_{A}_{τ}_{τ}_{ω} (1_{A}_{τ}_{A}

This is a contradiction.

(e) follows from (d) and Lemma 2.10.

Let (

(a) 0_{A}_{A}_{ω}

(b) The finite soft union of soft _{ω}_{ω}

(c) The arbitrary soft intersection of soft _{ω}_{ω}

(a) The proof follows from Theorems 2.2(a) and 2.4(b).

(b) We demonstrate that the soft union of two soft _{ω}_{ω}_{ω}_{θ}_{ω} (_{θ}_{ω} (

Hence, _{ω}

(c) Let _{λ}_{ω}_{λ}_{θ}_{ω} (_{λ}_{θ}_{ω} ( ∩̃ {_{λ}_{λ}_{x}_{θ}_{ω} ( ∩̃ {_{λ}_{x}_{τ}_{ω} (_{λ}_{A}_{τ}_{ω} (_{λ}_{A}_{x}_{θ}_{ω} (_{λ}_{λ}_{x}_{λ}

For any STS (_{θ}_{ω}

(1) According to Theorem 2.12(a), 0_{A}_{A}_{ω}_{A}_{A}_{θ}_{ω}.

(2) Let _{θ}_{ω}; then, 1_{A}_{A}_{ω}

Then, using Theorem 2.12(b), 1_{A}_{ω}_{θ}_{ω}.

(3) Let _{λ}_{θ}_{ω} for every _{A}_{λ}_{ω}

is soft _{ω}_{λ}_{θ}_{ω}.

Let (_{θ}_{ω} if and only if, for each _{x}_{x}_{τ}_{ω} (

Let _{θ}_{ω}, and let _{x}_{A}_{ω}_{x}_{A}_{A}_{A}_{ω}_{θ}_{ω} (1_{A}_{A}_{x}_{A}_{θ}_{ω} (1_{A}_{x}_{τ}_{ω} (_{A}_{A}_{x}_{τ}_{ω} (

Suppose that for each _{x}_{x}_{τ}_{ω} (_{θ}_{ω}. Then, _{θ}_{ω}(1_{A}_{A}_{x}_{θ}_{ω} (1_{A}_{A}_{x}_{τ}_{ω} (_{x}_{τ}_{ω} (_{A}_{A}_{x}_{A}_{θ}_{ω} (1_{A}

Every soft open, soft _{ω}

Let (_{x}_{τ}_{ω} (_{x}_{τ}_{ω} (_{ω}

Every countable soft open set in an STS is soft _{ω}

Using Theorem 2(d) of [2], countable soft sets in an STS are soft

For any STS (

(a) (

(b) _{θ}_{ω}.

(c) For every _{θ}_{ω} (_{τ}

(a) ⇒ (b): Suppose that (_{θ}_{ω}, let _{x}_{x}_{τ}_{ω}(_{θ}_{ω}. However, from Theorem 2.9, we obtain _{θ}_{ω} ⊆

(b) ⇒ (c): Suppose that _{θ}_{ω} and let _{θ}_{ω} (_{τ}_{x}_{A}_{τ}_{A}_{τ}_{A}_{τ}_{θ}_{ω}. Thus, according to Theorem 2.14, there exists _{x}_{τ}_{ω} (_{A}_{τ}_{x}_{τ}_{ω} (_{τ}_{ω} (_{τ}_{A}_{x}_{A}_{θ}_{ω} (_{τ}_{θ}_{ω} (

(c) ⇒ (a): Suppose that _{θ}_{ω} (_{τ}_{x}_{A}_{θ}_{ω} (1_{A}_{τ}_{A}_{A}_{x}_{A}_{θ}_{ω} (1_{A}_{x}_{τ}_{ω} (_{A}_{A}_{x}_{τ}_{ω} (

The inclusions in Theorem 2.9 are not equalities in general.

Let

Then,

and

Let (_{τ}_{a} (_{τ}

Let (_{θ}_{a}_{θ}

Suppose that _{θ}_{x}_{τ}_{a}_{τ}_{a} (_{τ}_{a}_{θ}

Let _{a}_{a}_{θ}

Suppose that

for all

Suppose that _{a}_{θ}_{x}_{a}_{θ}_{a}_{ℑ}_{a} (_{x}_{V}_{Cl}_{ℑa (}_{V}_{ )} ⊆̃

Let (_{θ}_{θ}

For each _{a}

Let (_{θ}_{ω}, _{a}_{θ}_{ω} for all

Suppose that _{θ}_{ω} and let _{x}_{θ}_{ω}, there exists _{x}_{τ}_{ω} (_{a}_{(}_{τ}_{a}_{)}_{ω,} (_{(}_{τ}_{ω}_{)}_{a} (_{τ}_{ω} (_{a}_{θ}_{ω}.

Let _{a}_{a}_{θ}_{ω} for all

Suppose that

Suppose that _{a}_{θ}_{ω} for all _{x}_{a}_{θ}_{ω}. Thus, there exists _{a}_{(ℑ}_{a}_{)}_{ω} (_{x}_{V}_{Cl}_{(ℑa)ω}_{(}_{V}_{)} ⊆̃

Let (_{θ}_{ω} if and only if _{θ}_{ω} for all

For each _{a}

Let (_{θ}_{ω}, _{θ}_{ω} and _{θ}_{ω}.

Let _{x}_{y}_{(}_{x,y}_{)} ∊̃ _{θ}_{ω}, there exists _{(}_{x,y}_{)} ∊̃ _{(}_{τ}_{*}_{σ}_{)}_{ω} (_{(}_{x,y}_{)} ∊̃ _{(}_{x,y}_{)} ∊̃ _{τ}_{ω} (_{σ}_{ω} (_{(}_{τ}_{*}_{σ}_{)}_{ω} (_{(}_{τ}_{*}_{σ}_{)}_{ω} (

Thus, we obtain _{x}_{τ}_{ω} (_{y}_{τ}_{ω} (_{θ}_{ω} and _{θ}_{ω}.

Let (_{θ}_{ω} and _{θ}_{ω}. Is it true that _{θ}_{ω} ?

If _{pu}_{pu}_{ω}

Let

Define

For every soft closed set _{pu}_{pu}_{ω}_{A}_{pu}_{A}_{pu}

Let _{pu}_{pu}_{pu}_{ω}_{pu}_{θ}_{θ}_{ω}

Let _{θ}_{y}_{pu}_{x}_{y}_{pu}_{x}_{x}_{θ}_{x}_{τ}_{y}_{pu}_{x}_{pu}_{pu}_{τ}_{pu}_{pu}_{pu}_{pu}_{ω}_{pu}_{τ}_{τ}_{ω} (_{pu}_{pu}_{τ}_{pu}_{θ}_{ω}. It follows that _{pu}_{θ}_{θ}_{ω}

Let _{pu}_{pu}_{pu}_{ω}_{ω}_{pu}_{θ}_{ω}_{θ}_{ω}

Let _{θ}_{ω} and let _{y}_{pu}_{x}_{y}_{pu}_{x}_{x}_{θ}_{ω}, there exists _{x}_{τ}_{ω} (_{y}_{pu}_{x}_{pu}_{pu}_{τ}_{ω} (_{pu}_{pu}_{pu}_{ω}_{ω}_{pu}_{τ}_{ω} (_{τ}_{ω} (_{pu}_{pu}_{τ}_{ω} (_{pu}_{pu}_{θ}_{ω}.

Let _{pu}_{pu}_{pu}_{ω}_{ω}_{pu}_{θ}_{ω}_{θ}_{ω}

Let _{θ}_{ω} and let _{pu}_{x}_{pu}_{x}_{τ}_{ω} (_{pu}_{pu}_{ω}_{ω}_{pu}_{θ}_{ω}_{θ}_{ω}

In this section, we introduce and investigate soft _{ω}

A soft function _{pu}_{x}_{pu}_{x}_{x}_{pu}_{τ}_{σ}

Let _{pu}

Suppose that _{pu}_{pu}_{x}_{p}_{(}_{x}_{)} ∊̃ _{V}_{pu}_{x}_{pu}_{τ}_{(}_{ℑ}_{)}(_{τ}_{(ℵ)}(_{V}_{τ}_{(}_{ℑ}_{)}(_{pu}_{τ}_{(}_{ℑ}_{)}(_{τ}_{(ℵ)}(_{V}_{ℑ}(_{(}_{τ}_{(}_{ℑ}_{))}_{a} (_{τ}_{(}_{ℑ}_{)}(_{τ}_{(ℵ)}(_{V}_{Cl}_{ℵ}_{(}_{V}_{)}(_{ℵ}(_{ℑ}(_{ℵ}(

Suppose that _{x}_{pu}_{x}_{p}_{(}_{x}_{)} ∊̃ _{ℑ}(_{ℵ}(_{ℵ}(_{(}_{τ}_{(ℵ))}_{u (a)} (_{ℵ}(_{τ}_{(}_{ℑ}_{)} (_{S}_{Cl}_{ℑ}_{ (}_{S}_{)}, and thus, _{pu}_{τ}_{(}_{ℑ}_{)}(_{S}_{pu}_{Cl}_{ℑ}_{ (}_{S}_{)}) = _{p}_{(}_{Cl}_{ℑ}_{ (}_{S}_{))}. Therefore, we obtain _{x}_{S}_{pu}_{τ}_{(}_{ℑ}_{)}(_{S}_{τ}_{(ℵ)}(_{pu}

A soft function _{pu}_{ω}_{x}_{pu}_{x}_{x}_{pu}_{τ}_{σ}_{ω} (

Let _{pu}_{ω}_{ω}

Suppose that _{pu}_{ω}_{pu}_{x}_{p}_{(}_{x}_{)} ∊̃ _{V}_{pu}_{ω}_{x}_{pu}_{τ}_{(}_{ℑ}_{)}(_{(}_{τ}_{(ℵ))}_{ω} (_{V}_{τ}_{(}_{ℑ}_{)}(_{pu}_{τ}_{(}_{ℑ}_{)}(_{τ}_{(ℵ)}(_{V}_{ℑ}(_{(}_{τ}_{(}_{ℑ}_{))}_{a} (_{τ}_{(}_{ℑ}_{)}(_{(}_{τ}_{(ℵ))}_{ω} (_{V}_{Cl}_{ℵω }_{(}_{V}_{)}(_{ℵ}_{ω} (_{ℑ}(_{ℵ}_{ω} (_{ω}

Suppose that _{ω}_{x}_{pu}_{x}_{p}_{(}_{x}_{)} ∊̃ _{ω}_{ℑ}(_{ℵ}_{ω} (_{ℵ}_{ω} (_{(}_{τ}_{(ℵ}_{ω}_{))}_{u(a)} (_{ℵ}_{ω} (_{τ}_{(}_{ℑ}_{)}(_{S}_{Cl}_{ℑ}_{ (}_{S}_{)}, and thus, _{pu}_{τ}_{(}_{ℑ}_{)}(_{S}_{pu}_{Cl}_{ℑ}_{ (}_{S}_{)}) = _{p}_{(}_{Cl}_{ℑ}_{ (}_{S}_{))}. Therefore, we obtain _{x}_{S}_{pu}_{τ}_{(}_{ℑ}_{)}(_{S}_{(}_{τ}_{(ℵ))}_{ω} (_{pu}_{ω}

Every soft _{ω}

Let _{pu}_{ω}_{x}_{pu}_{x}_{ω}_{pu}_{x}_{pu}_{τ}_{σ}_{ω} (_{σ}_{ω} (_{σ}_{pu}_{τ}_{σ}_{pu}

The converse of Theorem 3.5 is not true in general, as clarified by the following example:

Let _{ω}_{pu}_{ω}

If _{pu}_{pu}_{ω}

Suppose that _{pu}_{x}_{pu}_{x}_{pu}_{x}_{pu}_{τ}_{σ}_{σ}_{σ}_{ω} (_{pu}_{τ}_{σ}_{ω} (_{pu}_{ω}

Every soft continuous function is soft

The following two examples demonstrate that soft continuity and soft _{ω}

Let _{ω}_{pu}_{ω}

Let _{ω}_{pu}_{ω}

The following result provides a sufficient condition for a soft _{ω}

If _{pu}_{ω}_{pu}

Suppose that _{pu}_{ω}_{x}_{pu}_{x}_{pu}_{x}_{σ}_{ω} (_{pu}_{ω}_{x}_{pu}_{τ}_{σ}_{ω} (

Hence, _{pu}

Let _{pu}_{x}_{pu}_{x}_{θ}_{x}_{pu}_{σ}_{ω}(_{pu}_{ω}

Let _{x}_{pu}_{x}_{θ}_{x}_{pu}_{σ}_{ω} (_{x}_{θ}_{x}_{τ}_{x}_{pu}_{τ}_{pu}_{σ}_{ω}(_{pu}_{ω}

Let _{pu}_{pu}_{ω}

Using Theorem 3.12, it is sufficient to prove that for every _{x}_{pu}_{x}_{θ}_{x}_{pu}_{σ}_{ω} (_{x}_{pu}_{x}_{θ}_{pu}_{x}

If _{pu}_{ω}_{pu}_{θ}_{θ}_{ω} (_{pu}

Suppose that _{pu}_{ω}_{y}_{pu}_{θ}_{y}_{θ}_{ω} (_{pu}_{y}_{y}_{pu}_{θ}_{x}_{θ}_{y}_{pu}_{x}_{pu}_{ω}_{x}_{pu}_{τ}_{σ}_{ω} (_{x}_{x}_{θ}_{τ}_{A}_{pu}_{τ}_{pu}_{A}_{B}_{pu}_{τ}_{pu}_{τ}_{pu}_{σ}_{ω} (_{pu}_{σ}_{ω}(_{pu}_{B}_{y}_{θ}_{ω} (_{pu}

We have defined and investigated the _{ω}_{ω}_{ω}_{ω}

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

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E-mail: algore@just.edu.jo

International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(1): 89-99

**Published online** March 25, 2022 https://doi.org/10.5391/IJFIS.2022.22.1.89

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 (Samer Al Ghour)

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.

The soft *θ*_{ω}-closure operator is defined as a new soft operator that lies strictly between the usual soft closure and the soft *θ*-closure. Sufficient conditions are provided for equivalence between the soft *θ*_{ω}-closure and usual soft closure operators, and between the soft *θ*_{ω}-closure and soft *θ*-closure operators. Via the soft *θ*_{ω}-closure operator, the soft *θ*_{ω}-open sets are defined as a new class of soft sets that lies strictly between the class of soft open sets and the class of soft *θ*-open sets. It is proven that the class of soft *θ*_{ω}-open sets form a new soft topology. The soft ω-regularity is characterized via both the soft *θ*_{ω}-closure operator and soft *θ*_{ω}-open sets. The soft product theorem and several soft mapping theorems are introduced. The correspondence between the soft topology of the soft *θ*_{ω}-open sets of soft topological space and their generated topological spaces, and vice versa, are studied. In addition to these, soft *θ*_{ω}-continuity as a strong form of soft *θ*-continuity is introduced and investigated.

**Keywords**: Soft *&theta,*-closure, Soft *&theta,*-open, Soft &omega,-regular, Soft product, Soft *&theta,*-continuity, Soft generated soft topological space.

This paper follows the concepts and terminology that appear in [1–3]. In this paper, TS and STS denote the topological space and soft topological space, respectively. The concept of soft sets, which was introduced by Molodtsov [4] in 1999, is a general mathematical tool for dealing with uncertainty. Let _{A}_{A}_{A}_{A}_{A}_{ω}_{ω}_{ω}_{ω}_{ω}_{ω}_{ω}_{ω}_{ω}_{ω}_{ω}_{ω}

The following definitions and results are used throughout this work:

Let

(a) [1] _{Y}

(b) [1] _{Y}

(c) [46] _{x}

The set of all soft points in

Let _{x}_{x}_{x}_{x}_{x}

Let (_{a}

Let (

defines a soft topology on

Let _{a}

Then,

Let

for each (

Let (

Recall that a soft set

An STS (

(a) [50] soft locally indiscrete if every soft open set is soft closed;

(b) [2] soft locally countable if for each _{x}_{x}

(c) [2] soft anti-locally countable if for every _{A}

(d) [6] soft

(e) [6] soft _{x}_{A}_{ω}_{x}_{A}

In this section, we introduce the _{ω}_{ω}_{ω}_{ω}_{ω}_{ω}_{ω}_{ω}_{ω}

Let (

(a) A soft point _{x}_{x}_{θ}_{τ}_{A}_{x}

(b) _{θ}

(c) _{A}

(d) The family of all soft _{θ}

Let (

(a) (_{θ}

(b) _{θ}_{θ}

The following is the main definition of this work.

Let (

(a) A soft point _{x}_{ω}_{x}_{θ}_{ω} (_{τ}_{ω} (_{A}_{x}

(b) _{ω}_{θ}_{ω} (

(c) _{ω}_{A}_{ω}

(d) The family of all soft _{ω}_{θ}_{ω}.

Let (

(a) _{τ}_{θ}_{ω} (_{θ}

(b) If _{ω}

(c) If _{ω}

(a) To demonstrate that _{τ}_{θ}_{ω} (_{x}_{τ}_{x}_{x}_{τ}_{A}_{τ}_{ω} (_{τ}_{ω} (_{A}_{x}_{θ}_{ω} (_{θ}_{ω} (_{θ}_{x}_{θ}_{ω} (_{x}_{x}_{θ}_{ω} (_{τ}_{ω} (_{A}_{τ}_{ω} (_{τ}_{τ}_{A}_{x}_{θ}

(b) Suppose that _{θ}_{θ}_{ω} (_{ω}

(c) Suppose that _{ω}_{θ}_{ω} (_{τ}

Let (

(a) _{τ}_{θ}_{ω} (F).

(b) If _{ω}

(a) From Theorem 2.4 (a), _{τ}_{θ}_{ω} (_{θ}_{ω} (_{τ}_{x}_{θ}_{ω} (_{x}_{τ}_{ω} (_{A}_{τ}_{ω} (_{A}_{x}_{τ}

(b) Suppose that _{τ}_{θ}_{ω} (_{ω}

Let (

(a) _{τ}_{θ}_{ω} (

(b) If _{ω}

The proof follows from Theorem 2.5 and Theorem 7 of [6].

Let (

(a) _{τ}_{θ}_{ω} (

(b) If _{ω}

The proof follows from Theorem 2.5 and Theorem 6 of [6].

Let (

(a) _{θ}_{θ}_{ω} (

(b) If _{ω}

(a) Using Theorem 2.4(a), _{θ}_{ω} (_{θ}_{θ}_{θ}_{ω} (_{x}_{θ}_{x}_{τ}_{A}_{τ}_{ω} (_{τ}_{τ}_{ω} (_{A}_{x}_{θ}_{ω} (

(b) Suppose that _{ω}_{θ}_{ω} (_{θ}

For any STS (_{θ}_{θ}_{ω} ⊆

Let _{θ}_{A}_{A}_{ω}_{θ}_{ω}. Therefore, _{θ}_{θ}_{ω}. To demonstrate that _{θ}_{ω} ⊆ _{θ}_{ω}; then, 1_{A}_{ω}_{A}

Let (_{θ}_{τ}

Let _{τ}_{θ}_{θ}_{τ}_{x}_{θ}_{x}_{τ}_{A}_{y}_{τ}_{A}_{τ}_{θ}

Let (

(a) If _{A}_{θ}_{ω} (_{θ}_{ω} (

(b) For each _{θ}_{ω} (_{θ}_{ω} (_{θ}_{ω} (

(c) For each _{θ}_{ω} (

(d) For each _{ω}_{θ}_{ω} (_{τ}

(e) For each _{θ}_{θ}_{ω} (_{τ}

(a) Let _{x}_{θ}_{ω} (_{x}_{x}_{θ}_{ω} (_{τ}_{ω} (_{A}_{τ}_{ω} (_{A}_{x}_{θ}_{ω} (

(b) As _{θ}_{ω} (_{θ}_{ω} (_{θ}_{ω} (_{x}_{A}_{θ}_{ω} (_{θ}_{ω} (_{x}_{τ}_{ω} (_{A}_{τ}_{ω} (_{A}_{x}

Hence, _{x}_{A}_{θ}_{ω} (

(c) Let _{A}_{θ}_{ω} (_{x}_{A}_{θ}_{ω} (_{x}_{τ}_{ω} (_{A}_{θ}_{ω} (_{A}_{A}_{θ}_{ω} (

(d) Let _{ω}_{τ}_{θ}_{ω} (_{θ}_{ω} (_{τ}_{x}_{θ}_{ω} (_{A}_{τ}_{x}_{θ}_{ω}(_{x}_{A}_{τ}_{τ}_{ω} (1_{A}_{τ}_{A}

This is a contradiction.

(e) follows from (d) and Lemma 2.10.

Let (

(a) 0_{A}_{A}_{ω}

(b) The finite soft union of soft _{ω}_{ω}

(c) The arbitrary soft intersection of soft _{ω}_{ω}

(a) The proof follows from Theorems 2.2(a) and 2.4(b).

(b) We demonstrate that the soft union of two soft _{ω}_{ω}_{ω}_{θ}_{ω} (_{θ}_{ω} (

Hence, _{ω}

(c) Let _{λ}_{ω}_{λ}_{θ}_{ω} (_{λ}_{θ}_{ω} ( ∩̃ {_{λ}_{λ}_{x}_{θ}_{ω} ( ∩̃ {_{λ}_{x}_{τ}_{ω} (_{λ}_{A}_{τ}_{ω} (_{λ}_{A}_{x}_{θ}_{ω} (_{λ}_{λ}_{x}_{λ}

For any STS (_{θ}_{ω}

(1) According to Theorem 2.12(a), 0_{A}_{A}_{ω}_{A}_{A}_{θ}_{ω}.

(2) Let _{θ}_{ω}; then, 1_{A}_{A}_{ω}

Then, using Theorem 2.12(b), 1_{A}_{ω}_{θ}_{ω}.

(3) Let _{λ}_{θ}_{ω} for every _{A}_{λ}_{ω}

is soft _{ω}_{λ}_{θ}_{ω}.

Let (_{θ}_{ω} if and only if, for each _{x}_{x}_{τ}_{ω} (

Let _{θ}_{ω}, and let _{x}_{A}_{ω}_{x}_{A}_{A}_{A}_{ω}_{θ}_{ω} (1_{A}_{A}_{x}_{A}_{θ}_{ω} (1_{A}_{x}_{τ}_{ω} (_{A}_{A}_{x}_{τ}_{ω} (

Suppose that for each _{x}_{x}_{τ}_{ω} (_{θ}_{ω}. Then, _{θ}_{ω}(1_{A}_{A}_{x}_{θ}_{ω} (1_{A}_{A}_{x}_{τ}_{ω} (_{x}_{τ}_{ω} (_{A}_{A}_{x}_{A}_{θ}_{ω} (1_{A}

Every soft open, soft _{ω}

Let (_{x}_{τ}_{ω} (_{x}_{τ}_{ω} (_{ω}

Every countable soft open set in an STS is soft _{ω}

Using Theorem 2(d) of [2], countable soft sets in an STS are soft

For any STS (

(a) (

(b) _{θ}_{ω}.

(c) For every _{θ}_{ω} (_{τ}

(a) ⇒ (b): Suppose that (_{θ}_{ω}, let _{x}_{x}_{τ}_{ω}(_{θ}_{ω}. However, from Theorem 2.9, we obtain _{θ}_{ω} ⊆

(b) ⇒ (c): Suppose that _{θ}_{ω} and let _{θ}_{ω} (_{τ}_{x}_{A}_{τ}_{A}_{τ}_{A}_{τ}_{θ}_{ω}. Thus, according to Theorem 2.14, there exists _{x}_{τ}_{ω} (_{A}_{τ}_{x}_{τ}_{ω} (_{τ}_{ω} (_{τ}_{A}_{x}_{A}_{θ}_{ω} (_{τ}_{θ}_{ω} (

(c) ⇒ (a): Suppose that _{θ}_{ω} (_{τ}_{x}_{A}_{θ}_{ω} (1_{A}_{τ}_{A}_{A}_{x}_{A}_{θ}_{ω} (1_{A}_{x}_{τ}_{ω} (_{A}_{A}_{x}_{τ}_{ω} (

The inclusions in Theorem 2.9 are not equalities in general.

Let

Then,

and

Let (_{τ}_{a} (_{τ}

Let (_{θ}_{a}_{θ}

Suppose that _{θ}_{x}_{τ}_{a}_{τ}_{a} (_{τ}_{a}_{θ}

Let _{a}_{a}_{θ}

Suppose that

for all

Suppose that _{a}_{θ}_{x}_{a}_{θ}_{a}_{ℑ}_{a} (_{x}_{V}_{Cl}_{ℑa (}_{V}_{ )} ⊆̃

Let (_{θ}_{θ}

For each _{a}

Let (_{θ}_{ω}, _{a}_{θ}_{ω} for all

Suppose that _{θ}_{ω} and let _{x}_{θ}_{ω}, there exists _{x}_{τ}_{ω} (_{a}_{(}_{τ}_{a}_{)}_{ω,} (_{(}_{τ}_{ω}_{)}_{a} (_{τ}_{ω} (_{a}_{θ}_{ω}.

Let _{a}_{a}_{θ}_{ω} for all

Suppose that

Suppose that _{a}_{θ}_{ω} for all _{x}_{a}_{θ}_{ω}. Thus, there exists _{a}_{(ℑ}_{a}_{)}_{ω} (_{x}_{V}_{Cl}_{(ℑa)ω}_{(}_{V}_{)} ⊆̃

Let (_{θ}_{ω} if and only if _{θ}_{ω} for all

For each _{a}

Let (_{θ}_{ω}, _{θ}_{ω} and _{θ}_{ω}.

Let _{x}_{y}_{(}_{x,y}_{)} ∊̃ _{θ}_{ω}, there exists _{(}_{x,y}_{)} ∊̃ _{(}_{τ}_{*}_{σ}_{)}_{ω} (_{(}_{x,y}_{)} ∊̃ _{(}_{x,y}_{)} ∊̃ _{τ}_{ω} (_{σ}_{ω} (_{(}_{τ}_{*}_{σ}_{)}_{ω} (_{(}_{τ}_{*}_{σ}_{)}_{ω} (

Thus, we obtain _{x}_{τ}_{ω} (_{y}_{τ}_{ω} (_{θ}_{ω} and _{θ}_{ω}.

Let (_{θ}_{ω} and _{θ}_{ω}. Is it true that _{θ}_{ω} ?

If _{pu}_{pu}_{ω}

Let

Define

For every soft closed set _{pu}_{pu}_{ω}_{A}_{pu}_{A}_{pu}

Let _{pu}_{pu}_{pu}_{ω}_{pu}_{θ}_{θ}_{ω}

Let _{θ}_{y}_{pu}_{x}_{y}_{pu}_{x}_{x}_{θ}_{x}_{τ}_{y}_{pu}_{x}_{pu}_{pu}_{τ}_{pu}_{pu}_{pu}_{pu}_{ω}_{pu}_{τ}_{τ}_{ω} (_{pu}_{pu}_{τ}_{pu}_{θ}_{ω}. It follows that _{pu}_{θ}_{θ}_{ω}

Let _{pu}_{pu}_{pu}_{ω}_{ω}_{pu}_{θ}_{ω}_{θ}_{ω}

Let _{θ}_{ω} and let _{y}_{pu}_{x}_{y}_{pu}_{x}_{x}_{θ}_{ω}, there exists _{x}_{τ}_{ω} (_{y}_{pu}_{x}_{pu}_{pu}_{τ}_{ω} (_{pu}_{pu}_{pu}_{ω}_{ω}_{pu}_{τ}_{ω} (_{τ}_{ω} (_{pu}_{pu}_{τ}_{ω} (_{pu}_{pu}_{θ}_{ω}.

Let _{pu}_{pu}_{pu}_{ω}_{ω}_{pu}_{θ}_{ω}_{θ}_{ω}

Let _{θ}_{ω} and let _{pu}_{x}_{pu}_{x}_{τ}_{ω} (_{pu}_{pu}_{ω}_{ω}_{pu}_{θ}_{ω}_{θ}_{ω}

In this section, we introduce and investigate soft _{ω}

A soft function _{pu}_{x}_{pu}_{x}_{x}_{pu}_{τ}_{σ}

Let _{pu}

Suppose that _{pu}_{pu}_{x}_{p}_{(}_{x}_{)} ∊̃ _{V}_{pu}_{x}_{pu}_{τ}_{(}_{ℑ}_{)}(_{τ}_{(ℵ)}(_{V}_{τ}_{(}_{ℑ}_{)}(_{pu}_{τ}_{(}_{ℑ}_{)}(_{τ}_{(ℵ)}(_{V}_{ℑ}(_{(}_{τ}_{(}_{ℑ}_{))}_{a} (_{τ}_{(}_{ℑ}_{)}(_{τ}_{(ℵ)}(_{V}_{Cl}_{ℵ}_{(}_{V}_{)}(_{ℵ}(_{ℑ}(_{ℵ}(

Suppose that _{x}_{pu}_{x}_{p}_{(}_{x}_{)} ∊̃ _{ℑ}(_{ℵ}(_{ℵ}(_{(}_{τ}_{(ℵ))}_{u (a)} (_{ℵ}(_{τ}_{(}_{ℑ}_{)} (_{S}_{Cl}_{ℑ}_{ (}_{S}_{)}, and thus, _{pu}_{τ}_{(}_{ℑ}_{)}(_{S}_{pu}_{Cl}_{ℑ}_{ (}_{S}_{)}) = _{p}_{(}_{Cl}_{ℑ}_{ (}_{S}_{))}. Therefore, we obtain _{x}_{S}_{pu}_{τ}_{(}_{ℑ}_{)}(_{S}_{τ}_{(ℵ)}(_{pu}

A soft function _{pu}_{ω}_{x}_{pu}_{x}_{x}_{pu}_{τ}_{σ}_{ω} (

Let _{pu}_{ω}_{ω}

Suppose that _{pu}_{ω}_{pu}_{x}_{p}_{(}_{x}_{)} ∊̃ _{V}_{pu}_{ω}_{x}_{pu}_{τ}_{(}_{ℑ}_{)}(_{(}_{τ}_{(ℵ))}_{ω} (_{V}_{τ}_{(}_{ℑ}_{)}(_{pu}_{τ}_{(}_{ℑ}_{)}(_{τ}_{(ℵ)}(_{V}_{ℑ}(_{(}_{τ}_{(}_{ℑ}_{))}_{a} (_{τ}_{(}_{ℑ}_{)}(_{(}_{τ}_{(ℵ))}_{ω} (_{V}_{Cl}_{ℵω }_{(}_{V}_{)}(_{ℵ}_{ω} (_{ℑ}(_{ℵ}_{ω} (_{ω}

Suppose that _{ω}_{x}_{pu}_{x}_{p}_{(}_{x}_{)} ∊̃ _{ω}_{ℑ}(_{ℵ}_{ω} (_{ℵ}_{ω} (_{(}_{τ}_{(ℵ}_{ω}_{))}_{u(a)} (_{ℵ}_{ω} (_{τ}_{(}_{ℑ}_{)}(_{S}_{Cl}_{ℑ}_{ (}_{S}_{)}, and thus, _{pu}_{τ}_{(}_{ℑ}_{)}(_{S}_{pu}_{Cl}_{ℑ}_{ (}_{S}_{)}) = _{p}_{(}_{Cl}_{ℑ}_{ (}_{S}_{))}. Therefore, we obtain _{x}_{S}_{pu}_{τ}_{(}_{ℑ}_{)}(_{S}_{(}_{τ}_{(ℵ))}_{ω} (_{pu}_{ω}

Every soft _{ω}

Let _{pu}_{ω}_{x}_{pu}_{x}_{ω}_{pu}_{x}_{pu}_{τ}_{σ}_{ω} (_{σ}_{ω} (_{σ}_{pu}_{τ}_{σ}_{pu}

The converse of Theorem 3.5 is not true in general, as clarified by the following example:

Let _{ω}_{pu}_{ω}

If _{pu}_{pu}_{ω}

Suppose that _{pu}_{x}_{pu}_{x}_{pu}_{x}_{pu}_{τ}_{σ}_{σ}_{σ}_{ω} (_{pu}_{τ}_{σ}_{ω} (_{pu}_{ω}

Every soft continuous function is soft

The following two examples demonstrate that soft continuity and soft _{ω}

Let _{ω}_{pu}_{ω}

Let _{ω}_{pu}_{ω}

The following result provides a sufficient condition for a soft _{ω}

If _{pu}_{ω}_{pu}

Suppose that _{pu}_{ω}_{x}_{pu}_{x}_{pu}_{x}_{σ}_{ω} (_{pu}_{ω}_{x}_{pu}_{τ}_{σ}_{ω} (

Hence, _{pu}

Let _{pu}_{x}_{pu}_{x}_{θ}_{x}_{pu}_{σ}_{ω}(_{pu}_{ω}

Let _{x}_{pu}_{x}_{θ}_{x}_{pu}_{σ}_{ω} (_{x}_{θ}_{x}_{τ}_{x}_{pu}_{τ}_{pu}_{σ}_{ω}(_{pu}_{ω}

Let _{pu}_{pu}_{ω}

Using Theorem 3.12, it is sufficient to prove that for every _{x}_{pu}_{x}_{θ}_{x}_{pu}_{σ}_{ω} (_{x}_{pu}_{x}_{θ}_{pu}_{x}

If _{pu}_{ω}_{pu}_{θ}_{θ}_{ω} (_{pu}

Suppose that _{pu}_{ω}_{y}_{pu}_{θ}_{y}_{θ}_{ω} (_{pu}_{y}_{y}_{pu}_{θ}_{x}_{θ}_{y}_{pu}_{x}_{pu}_{ω}_{x}_{pu}_{τ}_{σ}_{ω} (_{x}_{x}_{θ}_{τ}_{A}_{pu}_{τ}_{pu}_{A}_{B}_{pu}_{τ}_{pu}_{τ}_{pu}_{σ}_{ω} (_{pu}_{σ}_{ω}(_{pu}_{B}_{y}_{θ}_{ω} (_{pu}

We have defined and investigated the _{ω}_{ω}_{ω}_{ω}

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