Article Search
닫기

## Original Article

Split Viewer

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

## Soft -Open Sets and Soft -Continuity

Samer Al Ghour

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

Correspondence to :
Samer Al Ghour (Samer Al Ghour)

Received: September 4, 2021; Revised: October 19, 2021; Accepted: November 2, 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 noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

The soft θω-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 θω-closure and usual soft closure operators, and between the soft θω-closure and soft θ-closure operators. Via the soft θω-closure operator, the soft θω-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 θω-open sets form a new soft topology. The soft ω-regularity is characterized via both the soft θω-closure operator and soft θω-open sets. The soft product theorem and several soft mapping theorems are introduced. The correspondence between the soft topology of the soft θω-open sets of soft topological space and their generated topological spaces, and vice versa, are studied. In addition to these, soft θω-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.

### 1. Introduction

This paper follows the concepts and terminology that appear in [13]. 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 X be a universal set and A be a set of parameters. A soft set over X relative to A is a function G: A(X). The family of all soft sets over X relative to A is denoted by SS (X, A). In this paper, the null soft set and absolute soft set are denoted by 0A and 1A, respectively. As a contemporary structure of mathematics, STSs were defined in [5] as follows. An STS is a triplet (X, τ, A), where τSS (X, A), τ contains 0A and 1A, τ is closed under a finite soft intersection, and τ is closed under an arbitrary soft union. Let (X, τ, A) be an STS and FSS(X, A); then, F is said to be a soft open set in (X, τ, A) if Fτ, and F is said to be a soft closed set in (X, τ, A) if 1AF is a soft open set in (X, τ, A). The concept of soft topology and its applications remain a hot research area [1,2,630]. The notions of the θ-closure operator and θ-open sets in TSs were introduced in [31]. Research relating to the θ-closure operator and θ-open sets remains an important field [3239]. In recent years, the θω-closure operator and θω-open sets in TSs were defined in [3], and their research was continued in [4043]. The notions of the soft θ-closure operator and soft θ-open sets in STSs were defined in [44]. In this work, the soft θω-closure operator is defined as a new soft operator that lies strictly between the usual soft closure and the soft θ-closure. Several sufficient conditions for equivalence between the soft θω-closure and usual soft closure operators, and between the soft θω-closure and soft sets that lies soft θ-closure operators, soft θ-closure operators, are provided. Via the soft θω-closure operator, the soft θω-open sets are defined as a new class of strictly between the class of soft open sets and the class of soft θ-open sets. It is proven that the class of soft θω-open sets forms a new soft topology. The soft ω-regularity is characterized through both the soft θω-closure operator and soft θω-open sets. The soft product theorem and several soft mapping theorems are introduced. The study of the relationships between ordinary TSs and STSs is an important research direction [45]. In this study, the correspondence between the soft topology of the soft θω-open sets of STS and their generated TSs, and vice versa, are studied. In addition to these, the soft θω-continuity is introduced and investigated as a strong form of soft θ-continuity. In future work, we hope to determine an application for our new soft concepts in a decision-making problem.

The following definitions and results are used throughout this work:

### Definition 1.1

Let X be a universal set and A be a set of parameters. Then, GSS(X,A) is defined by

(a) [1] G(a)={Y,if a=e,,if ae is denoted by eY.

(b) [1] G(a) = Y for all aA is denoted by CY.

(c) [46] G(a)={{x},if a=e,,if ae is denoted by ex and is known as a soft point.

The set of all soft points in SS(X,A) is denoted as SP (X,A).

### Definition 1.2 [46]

Let GSS(X,A) and axSP(X, A). Then, ax is said to belong to F (notation: ax ∊̃ G) if ax ⊆̃ G, or equivalently, ax ∊̃ G if and only if xG(a).

### Theorem 1.3 [5]

Let (X, τ,A) be an STS. Then, the collection {F(a): Fτ} defines a topology on X for every aA. This topology is denoted by τa.

### Theorem 1.4 [47]

Let (X, ℑ) be a TS. Then, the collection

{FSS(X,A):F(a)for all aA}

defines a soft topology on X relative to A. This soft topology is denoted by τ (ℑ).

### Theorem 1.5 [1]

Let X be an initial universe and let A be a set of parameters. Let {ℑa: aA} be an indexed family of topologies on X and let

τ={FSS(X,A):F(a)afor all aA}.

Then, τ defines a soft topology on X relative to A. This soft topology is denoted by aAa.

### Definition 1.6 [49]

Let FSS (X,A) and GSS (Y,B). Then, the soft Cartesian product of F and G is a soft set denoted by F × GSS (X × Y,A × B), and it is defined by

(F×G)((a,b))=F(a)×G(b),

for each (a, b) ∈ A × B.

### Definition 1.7 [50]

Let (X, τ,A) and (Y, σ,B) be two STSs, and let = {F ×G: Fτ and Gσ}. Subsequently, the soft topology on X × Y relative to A × B that has as a soft base is known as the product soft topology and is denoted by τ * σ.

Recall that a soft set GSS(X,A) is said to be a countable soft set if G(a) is a countable subset of X for every aA.

### Definition 1.8

An STS (X, τ,A) is said to be:

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

(b) [2] soft locally countable if for each axSP(X,A), there exists Gτ such that ax ∊̃ G, and G is a countable soft set;

(c) [2] soft anti-locally countable if for every Fτ − {0A}, F is not a countable soft set;

(d) [6] soft ω-locally indiscrete if every soft open set is soft ω-closed; and

(e) [6] soft ω-regular if, when M is soft closed and ax ∊̃ 1AM, there exist Fτ and Gτω such that ax ∊̃ F, M ⊆̃ G, and F ∩̃ G = 0A.

### 2. Soft θω-Closure Operator

In this section, we introduce the θω-closure operator that lies strictly between the usual soft closure and soft θ-closure. We provide sufficient conditions for equivalence between the soft θω-closure and usual soft closure operators, and between the soft θω-closure and soft θ-closure operators. We use the soft θω-closure operator to define soft θω-open sets as a new class of soft sets that lies strictly between the class of soft open sets and class of soft θ-open sets. We also demonstrate that the class of soft θω-open sets forms a soft topology. Moreover, we use the soft θω-closure operator and soft θω-open sets to characterize the soft ω-regularity. We also provide a soft product theorem and several soft mapping theorems. Finally, we study the correspondence between the soft topology of the soft θω-open sets of a STS and their generated TSs, and vice versa.

### Definition 2.1 [44]

Let (X, τ,A) be an STS and let FSS(X,A).

(a) A soft point axSP(X,A) is in the soft θ-closure of F (ax ∊̃ Clθ(F)) if Clτ (G) ∩̃ F ≠ 0A for any Gτ with ax ∊̃ G.

(b) F is soft θ-closed if Clθ(F) = F.

(c) F is soft θ-open if 1AF is θ-closed.

(d) The family of all soft θ-open sets in (X, τ,A) is denoted by τθ.

### Theorem 2.2 [44]

Let (X, τ,A) be an STS. Then:

(a) (X, τθ,A) is an STS.

(b) τθτ and τθτ in general.

The following is the main definition of this work.

### Definition 2.3

Let (X, τ,A) be an STS and let FSS(X, A).

(a) A soft point axSP(X,A) is in the soft θω-closure of F (ax ∊̃ Clθω (F),) if Clτω (G) ∩̃ F ≠ 0A for any Gτ with ax ∊̃ G.

(b) F is known as soft θω-closed if Clθω (F) = F.

(c) F is known as soft θω-open if 1AF is θω-closed.

(d) The family of all soft θω-open sets in (X, τ,A) is denoted by τθω.

### Theorem 2.4

Let (X, τ,A) be an STS and let FSS(X, A). Then:

(a) Clτ (F) ⊆̃ Clθω (F) ⊆̃ Clθ(F).

(b) If F is soft θ-closed, F is soft θω-closed.

(c) If F is soft θω-closed, F is soft closed.

Proof

(a) To demonstrate that Clτ (F) ⊆̃ Clθω (F), let ax ∊̃ Clτ (F) and let Gτ such that ax ∊̃ G. As ax ∊̃ Clτ (F), G ∩̃ F ≠ 0A. Because G ⊆̃ Clτω (G), Clτω (G) ∩̃ F ≠ 0A. Thus, ax ∊̃ Clθω (F). To demonstrate that Clθω (F) ⊆̃ Clθ(F), let ax ∊̃ Clθω (F) and let Gτ such that ax ∊̃ G. As ax ∊̃ Clθω (F), Clτω (G) ∩̃ F ≠ 0A. Because Clτω (G) ⊆̃ Clτ (G), Clτ (G) ∩̃ F ≠ 0A. Thus, ax ∊̃ Clθ(F).

(b) Suppose that F is soft θ-closed. Then, Clθ (F) = F, and according to (a), Clθω (F) = F. Hence, F is soft θω-closed.

(c) Suppose that F is soft θω-closed. Then, Clθω (F) = F, and according to (a), Clτ (F) = F. Hence, F is soft closed.

### Theorem 2.5

Let (X, τ,A) be soft ω-locally indiscrete, and let FSS(X,A). Then:

(a) Clτ (F) = Clθω (F).

(b) If F is soft closed, F is soft θω-closed.

Proof

(a) From Theorem 2.4 (a), Clτ (F) ⊆̃ Clθω (F). To show that Clθω (F) ⊆̃ Clτ (F), let ax ∊̃ Clθω (F) and Gτ, with ax ∊̃ G. Then, Clτω (G) ∩̃ F ≠ 0A. Because (X, τ,A) is soft ω-locally indiscrete, Clτω (G) = G; hence, G ∩̃ F ≠ 0A. Therefore, ax ∊̃ Clτ (F).

(b) Suppose that F is soft closed. Then, F = Clτ (F), and according to (a), F = Clθω (F). Hence, F is soft θω-closed.

### Corollary 2.6

Let (X, τ,A) be soft locally indiscrete and let FSS(X,A). Then:

(a) Clτ (F) = Clθω (F).

(b) If F is soft closed, F is soft θω-closed.

Proof

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

### Corollary 2.7

Let (X, τ,A) be soft locally countable and let FSS(X,A). Then:

(a) Clτ (F) = Clθω (F).

(b) If F is soft closed, F is soft θω-closed.

Proof

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

### Theorem 2.8

Let (X, τ,A) be a soft anti-locally countable STS and let FSS(X,A). Then:

(a) Clθ(F) = Clθω (F).

(b) If F is soft θω-closed, F is soft θ-closed.

Proof

(a) Using Theorem 2.4(a), Clθω (F) ⊆̃ Clθ(F). To observe that Clθ(F) ⊆̃ Clθω (F), let ax ∊̃ Clθ(F) and Gτ, with ax ∊̃ G. Then, Clτ (G) ∩̃ F ≠ 0A. Because (X, τ,A) is soft anti-locally countable, according to Theorem 14 of [2], Clτω (G) = Clτ (G); hence, Clτω (G) ∩̃ F ≠ 0A. Therefore, ax ∊̃ Clθω (F).

(b) Suppose that F is soft θω-closed. Then, F = Clθω (F), and according to (a), F = Clθ(F). Hence, F is soft θ-closed.

### Theorem 2.9

For any STS (X, τ,A), τθτθωτ.

Proof

Let Fτθ; then, 1AF is soft θ-closed. Therefore, using Theorem 2.4(b), 1AF is soft θω-closed, and hence, Fτθω. Therefore, τθ. τθω. To demonstrate that τθωτ, let Fτθω; then, 1AF is soft θω-closed. Thus, according to Theorem 2.4(c), 1AF is soft closed. Hence, Fτ.

### Lemma 2.10

Let (X, τ,A) be an STS. Then, for each Fτ, Clθ(F) = Clτ (F).

Proof

Let Fτ. Then, using Theorem 2.4(a), Clτ (F) ⊆̃ Clθ(F). To observe that Clθ(F) ⊆̃ Clτ (F), let ax ∊̃ Clθ(F) and let Gτ, with ax ∊̃ G. Then, Clτ (G) ∩̃ F ≠ 0A. Select by ∊̃ Clτ (G) ∩̃ F. As Fτ, G ∩̃ F ≠ 0A. It follows that Clτ (F) ⊆̃ Clθ(F).

### Theorem 2.11

Let (X, τ, A) be an STS.

(a) If M ⊆̃ N ⊆̃ 1A, Clθω (M) ⊆̃ Clθω (N).

(b) For each M,NSS(X,A), Clθω (M ∪̃ N)= Clθω (M) ∪̃ Clθω (N).

(c) For each MSS(X,A), Clθω (M) is soft closed in (X, τ,A).

(d) For each Fτω, Clθω (F) = Clτ (F).

(e) For each Fτ, Clθ(F) = Clθω (F) = Clτ (F).

Proof

(a) Let ax ∊̃ Clθω (M) and Gτ with ax ∊̃ G. As ax ∊̃ Clθω (M), Clτω (G) ∩̃ M ≠ 0A. Because M ⊆̃ N, we obtain Clτω (G) ∩̃ N ≠ 0A. This demonstrates that ax ∊̃ Clθω (N).

(b) As M ⊆̃ M ∪̃ N and N ⊆̃ M ∪̃ N, according to (a), Clθω (M) ∪̃ Clθω (N) ⊆̃ Clθω (M ∪̃ N). Conversely, let ax ∊̃ 1A−(Clθω (M) ∪̃ Clθω (N)). Then, there exist G,Hτ such that ax ∊̃ G ∩̃ H, Clτω (G) ∩̃ M = 0A, and Clτω (G) ∩̃ N = 0A. Therefore, ax ∊̃ G ∩̃ Hτ and

Clθω(G˜H)˜(M˜N)=(Clθω(G˜H)˜M)˜(Clθω(G˜H)˜N)˜(Clθω(G)˜M)˜(Clθω(H)˜N)=0A˜0A=0A.

Hence, ax ∊̃ 1A − (Clθω (M ∪̃ N)).

(c) Let MSS(X,A). We show that 1AClθω (M) ∈ τ. Let ax ∊̃ 1AClθω (M). Then, Gτ exists such that ax ∊̃ G with Clτω (G) ∩̃ M = 0A; thus, G ∩̃ Clθω (M). = 0A. Hence, 1AClθω (M) ∈ τ.

(d) Let Fτω. Then, using Theorem 2.4(a), Clτ (F) ⊆̃ Clθω (F). To observe that Clθω (F) ⊆̃ Clτ (F), we assume to the contrary that there exists ax ∊̃ Clθω (F) ∩̃ (1AClτ (F)). Because ax ∊̃ Clθω(F) and ax ∊̃ (1AClτ (F)) ∈ τ. Then, Clτω (1AClτ (F)) ∩̃ F ≠ 0A. However,

Clτω(1A-Clτ(F)),˜F˜(1A-Clτ(F))˜F˜(1A-F)˜F=0A.

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

### Theorem 2.12

Let (X, τ,A) be an STS. Then:

(a) 0A and 1A are soft θω-closed sets.

(b) The finite soft union of soft θω-closed sets is soft θω-closed.

(c) The arbitrary soft intersection of soft θω-closed sets is soft θω-closed.

Proof

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

(b) We demonstrate that the soft union of two soft θω-closed sets is soft θω-closed. Let M and N be any two soft θω-closed sets. Then, Clθω (M) = M and Clθω (N) = N. Thus, according to Theorem 2.11(b),

Clθω(M˜N)=Clθ,ω(M)˜ClθωN)=M˜N.

Hence, M ∪̃ N is soft θω-closed.

(c) Let Fλ be soft θω-closed for all λ ∈ Γ. Then, for all λ ∈ Γ, Fλ = Clθω (Fλ). We show that Clθω ( ∩̃ {Fλ: λ ∈ Γ}) ⊆̃∩̃ {Fλ: λ ∈ Γ}. Let ax ∊̃ Clθω ( ∩̃ {Fλ: λ ∈ Γ}) and let Gτ, with ax ∊̃ G; then, Clτω (G) ∩̃ ( ∩̃ {Fλ: λ ∈ Γ}) ≠ 0A. Thus, for every λ ∈ Γ, Clτω (G) ∩̃ Fλ ≠ 0A. Hence, ax ∊̃ Clθω (Fλ) = Fλ for every λ ∈ Γ. It follows that ax ∊̃∩̃ {Fλ: λ ∈ Γ}.

### Theorem 2.13

For any STS (X, τ,A), (X, τθω,A) is an STS.

Proof

(1) According to Theorem 2.12(a), 0A and 1A are soft θω-closed sets, and thus, 0A, 1Aτθω.

(2) Let M,Nτθω; then, 1AM and 1AN are soft θω-closed sets. Note that

1A-(M˜N)=(1A-M)˜(1A-N).

Then, using Theorem 2.12(b), 1A − (M ∩̃ N) is soft θω-closed. Thus, M ∩̃ Nτθω.

(3) Let Fλτθω for every λ ∈ Γ. Then, {1AFλ: λ ∈ Γ} is a family of soft θω-closed sets. Thus, according to Theorem 2.12(c),

1A-˜{Fλ:λΓ}=˜{1A-Fλ:λΓ}

is soft θω-closed. Therefore, ∪̃ {Fλ: λ ∈ Γ} ∈ τθω.

### Theorem 2.14

Let (X, τ,A) be an STS and FSS(X,A). Then, Fτθω if and only if, for each ax ∊̃ F, there exists Gτ such that ax ∊̃ G ⊆̃ Clτω (G) ⊆̃ F.

Proof
Necessity

Let Fτθω, and let ax ∊̃ F. Then, we determine that 1AF is soft θω-closed and ax ∊̃ 1A − (1AF). As 1AF is soft θω-closed, Clθω (1AF) = 1AF, and thus, ax ∊̃ 1AClθω (1AF). Hence, there exists Gτ such that ax ∊̃ G and Clτω (G) ∩̃ (1AF) = 0A. Thus, ax ∊̃ G ⊆̃ Clτω (G) ⊆̃ F.

Sufficiency

Suppose that for each ax ∊̃ F, there exists Gτ such that ax ∊̃ G ⊆̃ Clτω (G) ⊆̃ F. Suppose to the contrary that Fτθω. Then, Clθω(1AF) ≠ 1AF, and thus, there exists ax ∊̃ Clθω (1AF) − (1AF). By assumption, there exists Gτ such that ax ∊̃ G ⊆̃ Clτω (G) ⊆̃ F, and thus, ax ∊̃ Gτ with Clτω (G) ∩̃ (1AF) = 0A. This implies that ax ∊̃ 1AClθω (1AF), which is a contradiction.

### Corollary 2.15

Every soft open, soft ω-closed set in an STS is soft θω-open.

Proof

Let (X, τ,A) be an STS, and let F be a soft open and soft ω-closed set. Let ax ∊̃ F. Because F is soft ω-closed, Clτω (F) = F. Select G = F; then, Gτ and ax ∊̃ G = Clτω (G) = F ⊆̃ F. Thus, according to Theorem 2.14, F is soft θω-open.

### Corollary 2.16

Every countable soft open set in an STS is soft θω-open.

Proof

Using Theorem 2(d) of [2], countable soft sets in an STS are soft ω-closed. Therefore, using Corollary 2.15, we obtain the result.

### Theorem 2.17

For any STS (X, τ,A), the following are equivalent:

(a) (X, τ,A) is soft ω-regular.

(b) τ = τθω.

(c) For every FSS(X,A), Clθω (F) = Clτ (F).

Proof

(a) ⇒ (b): Suppose that (X, τ,A) is soft ω-regular. To observe that ττθω, let Fτ and let ax ∊̃ F. Because (X, τ,A) is soft ω-regular, according to Theorem 13 of [6], there exists Gτ such that ax ∊̃ G ⊆̃ Clτω(G) ⊆̃ F. Therefore, using Theorem 2.14, we obtain Fτθω. However, from Theorem 2.9, we obtain τθωτ.

(b) ⇒ (c): Suppose that τ = τθω and let FSS(X,A). To verify that Clθω (F), ⊆̃ Clτ (F), let ax ∊̃ 1AClτ (F). Because 1AClτ (F) ∈ τ, using (b), 1AClτ (F) ∈ τθω. Thus, according to Theorem 2.14, there exists Gτ such that ax ∊̃ G ⊆̃ Clτω (G) ⊆̃ 1AClτ (F). Therefore, we obtain ax ∊̃ Gτ and Clτω (G) ∩̃ F ⊆̃ Clτω (G) ∩̃ Clτ (F) = 0A. Hence, ax ∊̃ 1AClθω (F). However, using Theorem 2.4(a), we obtain Clτ (F) ⊆̃ Clθω (F).

(c) ⇒ (a): Suppose that Clθω (F) = Clτ (F) for every FSS(X,A). Let Fτ and let ax ∊̃ F. Then, 1AF is soft closed and using (c), Clθω (1AF) = Clτ (1AF) = 1AF. Therefore, ax ∊̃ 1AClθω (1AF), and thus, there exists Gτ such that ax ∊̃ G and Clτω (G) ∩̃ (1AF) = 0A. Hence, ax ∊̃ G ⊆̃ Clτω (G) ⊆̃ F. Therefore, according to Theorem 13 of [6], (X, τ,A) is soft ω-regular.

### Remark

The inclusions in Theorem 2.9 are not equalities in general.

### Example 2.18

Let X = ℝ, A = {a, b}, and for all eA}.

Then,

τθω={FSS(X,A):F(e){,,}for all eA},

and

τθ={FSS(X,A):F(e){,}for all eA}.

### Lemma 2.19 [5]

Let (X, τ,A) be an STS and FSS(X, A). Then, for all aA, Clτa (F (a)) ⊆ (Clτ (F)) (a).

### Theorem 2.20

Let (X, τ,A) be an STS. If Fτθ, F(a) ∈ (τa)θ for all aA.

Proof

Suppose that Fτθ and let aA. Let xF(a) exists Gτ such that ax ∊̃ G ⊆̃ Clτ (G) ⊆̃ F. Thus, we obtain G(a) ∈ τa, and using Lemma 2.19, xG(a) ⊆ Clτa (G(a)) ⊆ (Clτ (G)) (a) ⊆ F(a). Therefore, F(a) ∈ (τa)θ.

### Theorem 2.21

Let X be an initial universe and let A be a set of parameters. Let {ℑa: aA} be an indexed family of topologies on X. Then, F(aAa)θ if and only if F(a) ∈ (ℑa)θ for all aA.

Proof
Necessity

Suppose that F(aAa)θ. Then, according to Theorem 2.20, F(a)((aAa)a)θ

for all aA. However, from Theorem 3.7 of [1], (aAa)a=a for all aA. This ends the proof.

Sufficiency

Suppose that F(a) ∈ (ℑa)θ for all aA. Let ax ∊̃ F; then, xF(a) ∈ (ℑa)θ. Therefore, there exists V ∈ ℑa such that xVCla (V ) ⊆ F(a). Thus, we obtain ax ∊̃ aV ⊆̃ aCla (V ) ⊆̃ F, with aVaAa. Furthermore, according to Proposition 2 of [6], aCla(V)=ClaAa(aV). Hence, F(aAa)θ.

### Corollary 2.22

Let (X, ℑ) be a TS and let A be a set of parameters. Then, F ∈ (τ (ℑ))θ if and only if F(a) ∈ ℑθ for all aA.

Proof

For each aA, set ℑa = ℑ. Then, τ()=aAa. Thus, using Theorem 2.21, we obtain the result.

### Theorem 2.23

Let (X, τ,A) be an STS. If Fτθω, F(a) ∈ (τa)θω for all aA.

Proof

Suppose that Fτθω and let aA. Let xF(a); then, ax ∊̃ F. As Fτθω, there exists Gτ such that ax ∊̃ G ⊆̃ Clτω (G) ⊆̃ F. Thus, we obtain G(a) ∈ τa, and according to Lemma 2.19 and Theorem 7 of [2], xG(a) ⊆ Cl(τa)ω, (G(a)) = Cl(τω)a (G(a)) ⊆ (Clτω (G)) (a) ⊆ F(a). Therefore, F(a) ∈ (τa)θω.

### Theorem 2.24

Let X be an initial universe and let A be a set of parameters. Let {ℑa: aA} be an indexed family of topologies on X. Then, F(aAa)θω if and only if F(a) ∈ (ℑa)θω for all aA.

Proof
Necessity

Suppose that F(aAa)θω. Then, using Theorem 2.23, F(a)((aAa)a)θω for all aA. However, according to Theorem 3.7 of [1], (aAa)a=a for all aA. This ends the proof.

Sufficiency

Suppose that F(a) ∈ (ℑa)θω for all aA. Let ax ∊̃ F; then, xF(a) ∈ (ℑa)θω. Thus, there exists V ∈ ℑa such that xVCl(ℑa)ω (V ) ⊆ F(a). Thus, we have ax ∊̃ aV ⊆̃ aCl(a)ω(V) ⊆̃ F with aVaAa. Moreover, according to Proposition 2 in [6], aCl(a)ω(V)=ClaA(a)ω(aV). Moreover, using Theorem 8 of [2], aA(a)ω=(aAa)ω. Hence, F(aAa)θω.

### Corollary 2.25

Let (X, ℑ) be a TS and let A be a set of parameters. Then, F ∈ (τ (ℑ))θω if and only if F(a) ∈ ℑθω for all aA.

Proof

For each aA, set ℑa = ℑ. Then, τ()=aAa. Thus, using Theorem 2.24, we obtain the result.

### Theorem 2.26

Let (X, τ, A) and (Y, σ, B) be two STSs. Let FSS(X,A) and GSS(Y,B). If F ×G ∈ (τ * σ)θω, Fτθω and Gσθω.

Proof

Let ax ∊̃ F and by ∊̃ G, then (a, b)(x,y) ∊̃ F × G. Because F × G ∈ (τ * σ)θω, there exists Mτ * σ such that (a, b)(x,y) ∊̃ M ⊆̃ Cl(τ*σ)ω (M) ⊆̃ F × G. Select Kτ and Nσ such that (a, b)(x,y) ∊̃ K × N ⊆̃ M. Then, according to Proposition 3(b) of [6], (a, b)(x,y) ∊̃ K × N ⊆̃ Clτω (K) × Clσω (N) ⊆̃ Cl(τ*σ)ω (K × N) ⊆̃ Cl(τ*σ)ω (M) ⊆̃ F × G.

Thus, we obtain ax ∊̃ K ⊆̃ Clτω (K) ⊆̃ F and by ∊̃ N ⊆̃ Clτω (N) ⊆̃ G. Therefore, Fτθω and Gσθω.

### Question 2.27

Let (X, τ,A) and (Y, σ,B) be two STSs. Let Fτθω and Gσθω. Is it true that F × G ∈ (τ * σ)θω ?

If fpu: (X, τ,A) → (Y, σ,B) is a soft closed function, it is clear that fpu: (X, τ,A) → (Y, σω,B) is soft closed. The following example shows that the converse is not true in general.

### Example 2.28

Let τ = {FSS(ℝ, {a}): F(a) = ∅︀ or ℝ − F(a) is finite}.

Define u: {a} → {a} and p: ℝ → ℝ using u(a) = a and

p(x)={x,if x,2,if x-.

For every soft closed set F of (ℝ, τ, {a}), (fpu(F)) (a) ⊆ ℤ. Thus, fpu: (X, τ, {a}) → (X, τω, {a}) is soft closed. Because 1A is soft closed in (X, τ, {a}) but fpu (1A) is not soft closed in (X, τ, {a}), fpu: (X, τ, {a}) → (X, τ, {a}) is not soft closed.

### Theorem 2.29

Let fpu: (X, τ,A) → (Y, σ,B) be a soft function. If fpu: (X, τ,A) → (Y, σ,B) is soft open and fpu: (X, τ,A) → (Y, σω,B) is soft closed, fpu: (X, τθ,A) → (Y, σθω,B) is soft open.

Proof

Let Gτθ and let by ∊̃ fpu (G). Select ax ∊̃ G such that by = fpu (ax). As ax ∊̃ Gτθ, there exists Mτ such that ax ∊̃ M ⊆̃ Clτ (M) ⊆̃ G. Therefore, by = fpu (ax) ∊̃ fpu(M) ⊆̃ fpu (Clτ (M)) ⊆̃ fpu (G). Because fpu: (X, τ,A) → (Y, σ,B) is soft open, fpu(M) ∈ σ. As fpu: (X, τ,A) → (Y, σω,B) is soft closed, fpu (Clτ (M)) is soft ω-closed in (Y, σ,B), and thus, Clτω (fpu(M)) ⊆̃ fpu (Clτ (M)). Therefore, fpu (G) ∈ σθω. It follows that fpu: (X, τθ,A) → (Y, σθω,B) is soft open.

### Theorem 2.30

Let fpu: (X, τ,A) → (Y, σ,B) be a soft function. If fpu: (X, τ,A) → (Y, σ,B) is soft open and fpu: (X, τω,A) → (Y, σω,B) is soft closed, fpu: (X, τθω,A) → (Y, σθω,B) is soft open.

Proof

Let Gτθω and let by ∊̃ fpu (G). Select ax ∊̃ G such that by = fpu (ax). As ax ∊̃ Gτθω, there exists Mτ such that ax ∊̃ M ⊆̃ Clτω (M) ⊆̃ G. Thus, by = fpu (ax) ∊̃ fpu(M) ⊆̃ fpu (Clτω (M)) ⊆̃ fpu (G). Because fpu: (X, τ,A) → (Y, σ, B) is soft open, f(M) ∈ σ. As fpu: (X, τω,A) → (Y, σω,B) is soft closed, fpu (Clτω (M)) is ω-closed in (Y, σ,B); therefore, Clτω (fpu(M)) ⊆̃ fpu(Clτω (M)) ⊆̃ fpu (G). It follows that fpu (G) ∈ σθω.

### Theorem 2.31

Let fpu: (X, τ,A) → (Y, σ,B) be a soft function. If both fpu: (X, τ,A) → (Y, σ,B) and fpu: (X, τω,A) → (Y, σω,B) are soft continuous, fpu: (X, τθω,A) → (Y, σθω,B) is soft continuous.

Proof

Let Hσθω and let ax˜fpu-1(H). Then, fpu (ax) ∊̃ H; thus, we determine Mσ such that fpu (ax) ∊̃ M ⊆̃ Clτω (M) ⊆̃ H. Therefore, ax˜fpu-1(M)˜fpu-1(Clτω(M))˜fpu-1(H). As fpu: (X, τ,A) → (Y, σ,B) is soft continuous, fpu-1(M)τ. Because fpu: (X, τω,A) → (Y, σω,B) is soft continuous, and fpu-1(Clτω(M)) is soft ω-closed in (X, τ,A); thus, Clτω(fpu-1(M))˜fpu-1(Clτω(M))˜fpu-1(H). Hence, fpu-1(H)τθω. It follows that fpu: (X, τθω,A) → (Y, σθω,B) is soft continuous.

### 3. Soft θω-Continuity

In this section, we introduce and investigate soft θω-continuity as a strong form of soft θ-continuity.

### Definition 3.1 [44]

A soft function fpu: (X, τ,A) → (Y, σ,B) is said to be soft θ-continuous if for every axSP(X,A) and every Gσ such that fpu(ax) ∊̃ G, there exists Hτ such that ax ∊̃ H and fpu(Clτ (H)) ⊆̃ Clσ(G).

### Theorem 3.2

Let p: (X, ℑ) → (Y, ℵ) be a function between two TSs and let u: AB be a function between two sets of parameters. Then, fpu: (X, τ(ℑ),A) → (Y, τ(ℵ),B) is soft θ-continuous if and only if p: (X, ℑ) → (Y, ℵ) is θ-continuous.

Proof
Necessity

Suppose that fpu: (X, τ(ℑ),A) → (Y, τ(ℵ),B) is soft θ-continuous. Let xX and V ∈ ℵ such that p(x) ∈ V. Select aA; then, fpu(ax) = u(a)p(x) ∊̃ u(a)V. As fpu: (X, τ(ℑ),A) → (Y, τ(ℵ),B) is soft θ-continuous, there exists Hτ such that ax ∊̃ H and fpu(Clτ()(H)) ⊆̃ Clτ(ℵ)(u(a)V ). Thus, p((Clτ()(H))(a)⊆(fpu(Clτ()(H))) (u(a)) ⊆ (Clτ(ℵ)(u(a)V ))(u(a)). According to Lemma 2.19, p(Cl(H(a))) = p(Cl(τ())a (H(a))) ⊆ p((Clτ()(H))(a)). However, using Proposition 2 of [6], (Clτ(ℵ)(u(a)V )) (u(a)) = u(a)Cl(V )(u(a)) = Cl(V ). Therefore, we obtain xH(a) ∈ ℑ and p(Cl(H(a))) ⊆ Cl(V ). Hence, p: (X, ℑ) → (Y, ℵ) is θ-continuous.

### Sufficiency

Suppose that p: (X, ℑ) → (Y, ℵ) is θ-continuous. Let axSP(X,A) and let Gτ (ℵ) such that fpu (ax) = u(a)p(x) ∊̃ G. Then, p(x) ∈ G(u(a)) ∈ ℵ. Because p: (X, ℑ) → (Y, ℵ) is θ-continuous, there exists S ∈ ℑ such that x ∊̃ S and p (Cl(S)) ⊆ Cl(G(u(a))). According to Lemma 2.19, Cl(G(u(a))) = Cl(τ(ℵ))u (a) (G(u(a))) ⊆ (Cl(G)) (u(a)). Furthermore, from Proposition 2 of [6], Clτ() (aS) = aCl (S), and thus, fpu(Clτ()(aS)) = fpu(aCl (S)) = u(a)p(Cl (S)). Therefore, we obtain ax ∊̃ aSτ (ℑ) and fpu (Clτ()(aS)) ⊆̃ Clτ(ℵ)(G). Hence, fpu: (X, τ( ℑ),A) → (Y, τ(ℵ),B) is soft θ-continuous.

### Definition 3.3

A soft function fpu: (X, τ,A) → (Y, σ,B) is said to be soft θω-continuous if for every axSP(X,A) and every Gσ such that fpu(ax) ∊̃ G, there exists Hτ such that ax ∊̃ H and fpu(Clτ (H)) ⊆̃ Clσω (G).

### Theorem 3.4

Let p: (X, ℑ) → (Y, ℵ) be a function between two TSs and let u: AB be a function between two sets of parameters. Then, fpu: (X, τ(ℑ),A) → (Y, τ(ℵ),B) is soft θω-continuous if and only if p: (X, ℑ) → (Y, ℵ), is θω-continuous.

Proof
Necessity

Suppose that fpu: (X, τ(ℑ),A) → (Y, τ(ℵ),B) is soft θω-continuous. Let xX and V ∈ ℵ such that p(x) ∈ V. Select aA. Then, fpu(ax) = u(a)p(x) ∊̃ u(a)V. As fpu: (X, τ(ℑ),A) → (Y, τ(ℵ),B) is soft θω-continuous, there exists Hτ such that ax ∊̃ H and fpu(Clτ()(H)) ⊆̃ Cl(τ(ℵ))ω (u(a)V ). Thus, p((Clτ()(H))(a)) ⊆ (fpu(Clτ()(H)))(u(a)) ⊆ (Clτ(ℵ)(u(a)V ))(u(a)). According to Lemma 2.19, p(Cl(H(a))) = p(Cl(τ())a (H(a))) ⊆ p((Clτ()(H))(a)). In contrast, using Proposition 2 of [6], (Cl(τ(ℵ))ω (u(a)V ))(u(a)) = u(a)Clω (V )(u(a)) = Clω (V ). Therefore, we obtain xH(a) ∈ ℑ and p(Cl(H(a))),⊆ Clω (V ). Hence, p: (X, ℑ) → (Y, ℵ) is θω-continuous.

### Sufficiency

Suppose that p: (X, ℑ) → (Y, ℵ) is θω-continuous. Let axSP(X,A) and let Gτ (ℵ) such that fpu (ax) = u (a)p(x) ∊̃ G. Then, p(x) ∈ G(u(a)) ∈ ℵ. Because p: (X, ℑ) → (Y, ℵ) is θω-continuous, there exists S ∈ ℑ such that x ∊̃ S and p(Cl(S)) ⊆ Clω (G(u(a))). According to Lemma 2.19, Clω (G(u(a))) = Cl(τ(ℵω))u(a) (G(u(a))) ⊆ (Clω (G)) (u(a)). Furthermore, from Proposition 2 of [6], Clτ()(aS) = aCl (S), and thus, fpu(Clτ()(aS)) = fpu(aCl (S)) = u(a)p(Cl (S)). Therefore, we obtain ax ∊̃ aSτ (ℑ) and fpu(Clτ()(aS)) ⊆̃ Cl(τ(ℵ))ω (G). Hence, fpu: (X, τ (ℑ),A) → (Y, τ (ℵ),B) is soft θω-continuous.

### Theorem 3.5

Every soft θω-continuous function is soft θ-continuous.

Proof

Let fpu: (X, τ,A) → (Y, σ,B) be soft θω-continuous. Let axSP(X,A). and Gσ such that fpu(ax) ∊̃ G. Based on the soft θω-continuity of fpu, there exists Hτ such that ax ∊̃ H and fpu(Clτ (H)) ⊆̃ Clσω (G). Because Clσω (G) ⊆̃ Clσ(G), fpu(Clτ (H)) ⊆̃ Clσ(G). Therefore, fpu is soft θ-continuous.

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

### Example 3.6

Let X = Y = ℕ, ℑ = {∅︀, X}, ℵ = {∅︀}, ∪{UY: YU is finite }, A = ℝ, and B = {b, c}. Consider the functions p: XY and u: AB, which are defined as p(x) = x for all xX, u(t) = b if and u(t) = c if . Then, as demonstrated in Example 2.4 of [40], p: (X, ℑ) → (Y, ℵ) is θ-continuous but not θω-continuous. Hence, according to Theorems 3.2 and 3.4, fpu: (X, τ(ℑ),A) → (Y, τ(ℵ),B) is soft θ-continuous but not soft θω-continuous.

### Theorem 3.7

If fpu: (X, τ,A) → (Y, σ,B) is a soft θ-continuous function with (Y, σ,B) in which is soft anti-locally countable, fpu: (X, τ,A) → (Y, σ,B) is soft θω-continuous.

Proof

Suppose that fpu: (X, τ,A) → (Y, σ,B) is a soft θ-continuous function. Let axSP(X,A) and let Gσ such that fpu(ax) ∊̃ G. Then, according to the soft θ-continuity of fpu, there exists Hτ such that ax ∊̃ H and fpu(Clτ (H)) ⊆̃ Clσ(G). As (Y, σ,B) is soft anti-locally countable, according to Theorem 14 of [2], Clσ(G) = Clσω (G), and thus, fpu(Clτ (H)) ⊆̃ Clσω (G). Therefore, fpu is soft θω-continuous.

### Theorem 3.8 [51]

Every soft continuous function is soft θ-continuous, but the converse is not true.

The following two examples demonstrate that soft continuity and soft θω-continuity are independent concepts:

### Example 3.9

Let X = Y = ℝ, ℑ = {∅︀, X}, ℵ = {∅︀}, ∪{UY: YU is finite }, A = ℝ, and B = {b, c}. Consider the functions p: XY and u: AB, which are defined as p(x) = x for all xX, u(t) = b if and u(t) = c if . Then, as proven in Example 2.7 of [40], p: (X, ℑ) → (Y, ℵ) is θω-continuous, but not continuous. Hence, according to Theorem 3.4 and Theorem 5.31 of [1], fpu: (X, τ(ℑ),, A) → (Y, τ(ℵ), B) is soft θω-continuous but not soft continuous.

### Example 3.10

Let X = Y = ℕ, ℑ = ℵ = {∅︀, X, {1}}, A = ℝ, and B = {b, c}. Consider the functions p: XY and u: AB, which are defined as p(x) = x for all xX, u(t) = b if and u(t) = c if . Then, as shown in Example 2.8 of [40], p: (X, ℑ) → (Y, ℵ) is continuous, but not θω-continuous. Hence, according to Theorem 5.31 of [1] and Theorem 3.4, fpu: (X, τ(ℑ),A) → (Y, τ(ℵ),B) is soft continuous, but not soft θω-continuous.

The following result provides a sufficient condition for a soft θω-continuous function to be soft continuous:

### Theorem 3.11

If fpu: (X, τ,A) → (Y, σ,B) is a soft θω-continuous function in which (Y, σ,B) is soft ω-regular, fpu: (X, τ,A) → (Y, σ,B) is soft continuous.

Proof

Suppose that fpu: (X, τ,A) → (Y, σ,B) is a soft θω-continuous function. Let axSP(X,A) and let Gσ such that fpu(ax) ∊̃ G. As (Y, σ,B) is soft ω-regular, there exists Hσ such that fpu(ax) ∊̃ H ⊆̃ Clσω (H) ⊆̃ G. Because fpu is soft θω-continuous, there exists Mτ such that ax ∊̃ M and fpu(Clτ (M)) ⊆̃ Clσω (H). Therefore, we obtain

f p u ( M ) ˜ f p u ( C l τ ( M ) ) ˜ C l σ ω ( H ) ˜ G

Hence, fpu is soft continuous.

### Theorem 3.12

Let fpu: (X, τ,A) → (Y, σ,B) be a soft function such that for every axSP(X,A) and every Gσ such that fpu(ax) ∊̃ G, there exists Hτθ such that ax ∊̃ H and fpu(H) ⊆̃ Clσω(G); then, fpu is soft θω-continuous.

Proof

Let axSP(X,A) and let Gσ such that fpu(ax) ∊̃ G. Then, by assumption, there exists Hτθ such that ax ∊̃ H and fpu(H) ⊆̃ Clσω (G). Because ax ∊̃ Hτθ, there exists Mτ such that ax ∊̃ M ⊆̃ Clτ (M) ⊆̃ H. Therefore, we obtain ax ∊̃ Mτ with fpu(Clτ (M)) ⊆̃ fpu(H) ⊆̃ Clσω(G). Hence, fpu: (X, τ,A) →.(Y, σ,B) is soft θω-continuous.

### Theorem 3.13

Let fpu: (X, τ,A) → (Y, σ,B) be a soft function such that for every Gσ, f p u - 1 ( C l σ ω ( G ) ) τ θ, fpu is soft θω-continuous.

Proof

Using Theorem 3.12, it is sufficient to prove that for every axSP(X,A) and every Gσ such that fpu(ax) ∊̃ G, there exists Hτθ such that ax ∊̃ H and fpu(H) ⊆̃ Clσω (G). Let axSP(X,A) and let Gσ such that fpu(ax) ∊̃ G. Set H = f p u - 1 ( C l σ ω ( G ) ). Then, by assumption, Hτθ. Furthermore, because fpu(ax) ∊̃ G, a x ˜ f p u - 1 ( G ) ˜ f p u - 1 ( C l σ ω ( G ) ) = H. Moreover, as H = f p u - 1 ( C l σ ω ( G ) ) , f p u ( H ) = f p u ( f p u - 1 ( C l σ ω ( G ) ) ) ˜ C l σ ω ( G ).

### Theorem 3.14

If fpu: (X, τ,A) → (Y, σ,B) is soft θω-continuous, for every FSS(X,A), fpu(Clθ(F)) ⊆̃ Clθω (fpu (F)).

Proof

Suppose that fpu: (X, τ,A) → (Y, σ,B) is soft θω-continuous and let FSS(X,A). Let by ∊̃ fpu (Clθ(F)). To observe that by ∊̃ Clθω (fpu(F)), let Gσ such that by ∊̃ G. Because by ∊̃ fpu (Clθ(F)), there exists ax ∊̃ Clθ(F) such that by = fpu (ax). Because fpu is soft θω-continuous, there exists Hτ such that ax ∊̃ H and fpu(Clτ (H)) ⊆̃ Clσω (G). As ax ∊̃ Hτ and ax ∊̃ Clθ(F), Clτ (H) ∩̃ F ≠ 0A, and thus, fpu(Clτ (H) ∩̃ F) ≠ fpu (0A) = 0B. Because fpu(Clτ (H) ∩̃ F) ⊆̃ fpu(Clτ(H)) ∩̃ fpu(F) ⊆̃ Clσω (G) ∩̃ fpu(F), Clσω(G) ∩̃ fpu(F) ≠ 0B. It follows that by ∊̃ Clθω (fpu(F)).

### 4. Conclusion

We have defined and investigated the θω-closure operator. Furthermore, we defined soft θω-open sets as a new class of soft sets that form a new soft topology. We have also introduced the soft θω-continuity as a strong form of θ-continuity. We have presented several characterizations, implications, and examples relating to these concepts. The following topics could be considered in future studies: solving an open question raised in this work, and defining soft separation axioms via soft θω-open sets.

### Conflict of Interest

1. Al Ghour, S, and Bin-Saadon, A (2019). On some generated soft topological spaces and soft homogeneity. Heliyon. 5. article no. e02061
2. Al Ghour, S, and Hamed, W (2020). On two classes of soft sets in soft topological spaces. Symmetry. 12. article no. 265
3. Al Ghour, S, and Irshedat, B (2017). The topology of θω-open sets. Filomat. 31, 5369-5377. https://doi.org/10.2298/FIL1716369A
4. Molodtsov, D (1999). Soft set theory—first results. Computers & Mathematics with Applications. 37, 19-31. https://doi.org/10.1016/S0898-1221(99)00056-5
5. Shabir, M, and Naz, M (2011). On soft topological spaces. Computers & Mathematics with Applications. 61, 1786-1799. https://doi.org/10.1016/j.camwa.2011.02.006
6. Al Ghour, S (2021). Weaker forms of soft regular and soft T2 soft topological spaces. Mathematics. 9. article no. 2153
7. Al Ghour, S (2021). Soft minimal soft sets and soft prehomogeneity in soft topological spaces. International Journal of Fuzzy Logic and Intelligent Systems. 21, 269-279. https://doi.org/10.5391/IJFIS.2021.21.3.269
8. Kocinac, LD, Al-shami, TM, and Cetkin, V (2021). Selection principles in the context of soft sets: Menger spaces. Soft Computing. 25, 12693-12702. https://doi.org/10.1007/s00500-021-06069-6
9. Al-shami, TM (2021). New soft structure: infra soft topological spaces. Mathematical Problems in Engineering. 2021. article no. 3361604
10. Al-shami, TM (2021). Infra soft compact spaces and application to fixed point theorem. Journal of Function Spaces. 2021. article no. 3417096
11. Asaad, BA, Al-shami, TM, and Mhemdi, A (2021). Bioperators on soft topological spaces. AIMS Mathematics. 6, 12471-12490. https://doi.org/10.3934/math.2021720
12. Al-shami, TM, and Mhemdi, A (2021). Belong and nonbelong relations on double-framed soft sets and their applications. Journal of Mathematics. 2021. article no. 9940301
13. Alkhazaleh, S, and Marei, EA (2021). New soft rough set approximations. International Journal of Fuzzy Logic and Intelligent Systems. 21, 123-134. https://doi.org/10.5391/IJFIS.2021.21.2.123
14. Al Ghour, S (2021). Strong form of soft semi-open sets in soft topological spaces. International Journal of Fuzzy Logic and Intelligent Systems. 21, 159-168. https://doi.org/10.5391/IJFIS.2021.21.2.159
15. Al Ghour, S (2021). Soft ω-paracompactness in soft topological spaces. International Journal of Fuzzy Logic and Intelligent Systems. 21, 57-65. https://doi.org/10.5391/IJFIS.2021.21.1.57
16. Al-shami, TM, and Abo-Tabl, ESA (2021). Connectedness and local connectedness on infra soft topological spaces. Mathematics. 9. article no. 1759
17. Mousarezaei, R, and Davvaz, B (2021). On soft topological polygroups and their examples. International Journal of Fuzzy Logic and Intelligent Systems. 21, 29-37. https://doi.org/10.5391/IJFIS.2021.21.1.29
18. Oztunc, S, Aslan, S, and Dutta, H (2021). Categorical structures of soft groups. Soft Computing. 25, 3059-3064. https://doi.org/10.1007/s00500-020-05362-0
19. Al-Shami, TM, and Abo-Tabl, EA (2021). Soft α-separation axioms and α-fixed soft points. AIMS Mathematics. 6, 5675-5694. https://doi.org/10.3934/math.2021335
20. Al-Shami, TM (2021). Bipolar soft sets: relations between them and ordinary points and their applications. Complexity. 2021. article no. 6621854
21. Al-Shami, TM, Alshammari, I, and Asaad, BA (2020). Soft maps via soft somewhere dense sets. Filomat. 34, 3429-3440. https://doi.org/10.2298/FIL2010429A
22. Oguz, G (2020). Soft topological transformation groups. Mathematics. 8. article no. 1545
23. Min, WK (2020). On soft ω-structures defined by soft sets. International Journal of Fuzzy Logic and Intelligent Systems. 20, 119-123. https://doi.org/10.5391/IJFIS.2020.20.2.119
24. Cetkin, V, Guner, E, and Aygun, H (2020). On 2S-metric spaces. Soft Computing. 24, 12731-12742. https://doi.org/10.1007/s00500-020-05134-w
25. El-Shafei, ME, and Al-Shami, TM (2020). Applications of partial belong and total non-belong relations on soft separation axioms and decision-making problem. Computational and Applied Mathematics. 39. article no. 138
26. Alcantud, JCR (2020). Soft open bases and a novel construction of soft topologies from bases for topologies. Mathematics. 8. article no. 672
27. Bahredar, AA, and Kouhestani, N (2020). On ∊-soft topological semigroups. Soft Computing. 24, 7035-7046. https://doi.org/10.1007/s00500-020-04826-7
28. Al-shami, TM, and El-Shafei, ME (2020). Partial belong relation on soft separation axioms and decision-making problem, two birds with one stone. Soft Computing. 24, 5377-5387. https://doi.org/10.1007/s00500-019-04295-7
29. Al-shami, TM, Kocinac, LD, and Asaad, BA (2020). Sum of soft topological spaces. Mathematics. 8. article no. 990
30. Al-Saadi, HS, and Min, WK (2017). On soft generalized closed sets in a soft topological space with a soft weak structure. International Journal of Fuzzy Logic and Intelligent Systems. 17, 323-328. https://doi.org/10.5391/IJFIS.2017.17.4.323
31. Velichko, NV (1966). H-closed topological spaces. Mat Sb. 70, 98-112.
32. Hassan, JA, and Labendia, MA (2022). θs-open sets and θscontinuity of maps in the product space. Journal of Mathematics and Computer Science. 25, 182-190. https://doi.org/10.22436/jmcs.025.02.07
33. Osipov, AV (2021). On the cardinality of S(n)-spaces. Quaestiones Mathematicae. 44, 121-128. https://doi.org/10.2989/16073606.2019.1672112
34. Periyasamy, P, and Ramesh, PR (2020). Local δ-closure functions in ideal topological spaces. Advances in Mathematics: Scientific Journal. 9, 2427-2436. https://doi.org/10.37418/amsj.9.5.6
35. Ramirez-Paramo, A (2019). A generalization of some cardinal function inequalities. Topology and its Applications. 256, 228-234. https://doi.org/10.1016/j.topol.2019.02.009
36. Babinkostova, L, Pansera, BA, and Scheepers, M (2019). Selective versions of θ-density. Topology and its Applications. 258, 268-281. https://doi.org/10.1016/j.topol.2019.02.061
37. Modak, S, and Noiri, T (2019). Some generalizations of locally closed sets. Iranian Journal of Mathematical Sciences and Informatics. 14, 159-165. https://doi.org/10.7508/ijmsi.2019.01.014
38. Reyes, JD, and Morales, AR (2018). The weak Urysohn number and upper bounds for cardinality of Hausdorff spaces. Houston Journal of Mathematics. 44, 1389-1398.
39. Al Ghour, S, and Al-Zoubi, S (2021). A new class between theta open sets and theta omega open sets. Heliyon. 7. article no. e05996
40. Al Ghour, S, and Irshidat, B (2020). On θω continuity. Heliyon. 6. article no. e03349
41. Butanas, LLL, and Labendia, MA (2020). θω-connected space and θω-continuity in the product space. Poincare Journal of Analysis and Applications. 7, 79-88. https://doi.org/10.46753/pjaa.2020.v07i01.008
42. Latif, RM (2020). Theta-ω-mappings in topological spaces. WSEAS Transactions on Mathematics. 19, 186-207. https://doi.org/10.37394/23206.2020.19.18
43. Al Ghour, S, and El-Issa, S (2019). θω-Connectedness and ω-R1 properties. Proyecciones (Journal of Mathematics). 38, 921-942. https://doi.org/10.22199/issn.0717-6279-2019-05-0059
44. Georgiou, DN, Megaritis, AC, and Petropoulos, VI (2013). On soft topological spaces. Applied Mathematics & Information Sciences. 7, 1889-1901. https://doi.org/10.12785/amis/070527
45. Al-shami, TM, and Kocinac, LD (2019). The equivalence between the enriched and extended soft topologies. Applied and Computational Mathematics. 18, 149-162.
46. Das, S, and Samanta, S (2013). Soft metric. Annals of Fuzzy Mathematics and Informatics. 6, 77-94.
47. Nazmul, S, and Samanta, SK (2013). Neighbourhood properties of soft topological spaces. Annals of Fuzzy Mathematics and Informatics. 6, 1-15.
48. Babitha, KV, and Sunil, J (2010). Soft set relations and functions. Computers & Mathematics with Applications. 60, 1840-1849. https://doi.org/10.1016/j.camwa.2010.07.014
49. Aygunoglu, A, and Aygun, H (2012). Some notes on soft topological spaces. Neural Computing and Applications. 21, 113-119. https://doi.org/10.1007/s00521-011-0722-3
50. Sayed, ME, and El-Bably, MK (2017). Soft simply open sets in soft topological space. Journal of Computational and Theoretical Nanoscience. 14, 4100-4103. https://doi.org/10.1166/jctn.2017.6792
51. Fomin, S (1943). Extensions of topological spaces. Annals of Mathematics. 43, 471-480. https://doi.org/10.2307/1968976

Samer Al Ghour received his Ph.D. degree in Mathematics from the University of Jordan, Jordan, in 1999. He is currently a professor at the Department of Mathematics and Statistics, Jordan University of Science and Technology, Jordan. His research interests include general topology, fuzzy topology, and soft set theory.

E-mail: algore@just.edu.jo

### Article

#### Original Article

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

## Soft -Open Sets and Soft -Continuity

Samer Al Ghour

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

Correspondence to:Samer Al Ghour (Samer Al Ghour)

Received: September 4, 2021; Revised: October 19, 2021; Accepted: November 2, 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 noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

The soft θω-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 θω-closure and usual soft closure operators, and between the soft θω-closure and soft θ-closure operators. Via the soft θω-closure operator, the soft θω-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 θω-open sets form a new soft topology. The soft ω-regularity is characterized via both the soft θω-closure operator and soft θω-open sets. The soft product theorem and several soft mapping theorems are introduced. The correspondence between the soft topology of the soft θω-open sets of soft topological space and their generated topological spaces, and vice versa, are studied. In addition to these, soft θω-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.

### 1. Introduction

This paper follows the concepts and terminology that appear in [13]. 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 X be a universal set and A be a set of parameters. A soft set over X relative to A is a function G: A(X). The family of all soft sets over X relative to A is denoted by SS (X, A). In this paper, the null soft set and absolute soft set are denoted by 0A and 1A, respectively. As a contemporary structure of mathematics, STSs were defined in [5] as follows. An STS is a triplet (X, τ, A), where τSS (X, A), τ contains 0A and 1A, τ is closed under a finite soft intersection, and τ is closed under an arbitrary soft union. Let (X, τ, A) be an STS and FSS(X, A); then, F is said to be a soft open set in (X, τ, A) if Fτ, and F is said to be a soft closed set in (X, τ, A) if 1AF is a soft open set in (X, τ, A). The concept of soft topology and its applications remain a hot research area [1,2,630]. The notions of the θ-closure operator and θ-open sets in TSs were introduced in [31]. Research relating to the θ-closure operator and θ-open sets remains an important field [3239]. In recent years, the θω-closure operator and θω-open sets in TSs were defined in [3], and their research was continued in [4043]. The notions of the soft θ-closure operator and soft θ-open sets in STSs were defined in [44]. In this work, the soft θω-closure operator is defined as a new soft operator that lies strictly between the usual soft closure and the soft θ-closure. Several sufficient conditions for equivalence between the soft θω-closure and usual soft closure operators, and between the soft θω-closure and soft sets that lies soft θ-closure operators, soft θ-closure operators, are provided. Via the soft θω-closure operator, the soft θω-open sets are defined as a new class of strictly between the class of soft open sets and the class of soft θ-open sets. It is proven that the class of soft θω-open sets forms a new soft topology. The soft ω-regularity is characterized through both the soft θω-closure operator and soft θω-open sets. The soft product theorem and several soft mapping theorems are introduced. The study of the relationships between ordinary TSs and STSs is an important research direction [45]. In this study, the correspondence between the soft topology of the soft θω-open sets of STS and their generated TSs, and vice versa, are studied. In addition to these, the soft θω-continuity is introduced and investigated as a strong form of soft θ-continuity. In future work, we hope to determine an application for our new soft concepts in a decision-making problem.

The following definitions and results are used throughout this work:

### Definition 1.1

Let X be a universal set and A be a set of parameters. Then, GSS(X,A) is defined by

(a) [1] $G(a)={Y,if a=e,∅,if a≠e$ is denoted by eY.

(b) [1] G(a) = Y for all aA is denoted by CY.

(c) [46] $G(a)={{x},if a=e,∅,if a≠e$ is denoted by ex and is known as a soft point.

The set of all soft points in SS(X,A) is denoted as SP (X,A).

### Definition 1.2 [46]

Let GSS(X,A) and axSP(X, A). Then, ax is said to belong to F (notation: ax ∊̃ G) if ax ⊆̃ G, or equivalently, ax ∊̃ G if and only if xG(a).

### Theorem 1.3 [5]

Let (X, τ,A) be an STS. Then, the collection {F(a): Fτ} defines a topology on X for every aA. This topology is denoted by τa.

### Theorem 1.4 [47]

Let (X, ℑ) be a TS. Then, the collection

${F∈SS(X,A):F(a)∈ℑ for all a∈A}$

defines a soft topology on X relative to A. This soft topology is denoted by τ (ℑ).

### Theorem 1.5 [1]

Let X be an initial universe and let A be a set of parameters. Let {ℑa: aA} be an indexed family of topologies on X and let

$τ={F∈SS(X,A):F(a)∈ℑa for all a∈A}.$

Then, τ defines a soft topology on X relative to A. This soft topology is denoted by $⊕a∈Aℑa$.

### Definition 1.6 [49]

Let FSS (X,A) and GSS (Y,B). Then, the soft Cartesian product of F and G is a soft set denoted by F × GSS (X × Y,A × B), and it is defined by

$(F×G)((a,b))=F(a)×G(b),$

for each (a, b) ∈ A × B.

### Definition 1.7 [50]

Let (X, τ,A) and (Y, σ,B) be two STSs, and let = {F ×G: Fτ and Gσ}. Subsequently, the soft topology on X × Y relative to A × B that has as a soft base is known as the product soft topology and is denoted by τ * σ.

Recall that a soft set GSS(X,A) is said to be a countable soft set if G(a) is a countable subset of X for every aA.

### Definition 1.8

An STS (X, τ,A) is said to be:

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

(b) [2] soft locally countable if for each axSP(X,A), there exists Gτ such that ax ∊̃ G, and G is a countable soft set;

(c) [2] soft anti-locally countable if for every Fτ − {0A}, F is not a countable soft set;

(d) [6] soft ω-locally indiscrete if every soft open set is soft ω-closed; and

(e) [6] soft ω-regular if, when M is soft closed and ax ∊̃ 1AM, there exist Fτ and Gτω such that ax ∊̃ F, M ⊆̃ G, and F ∩̃ G = 0A.

### 2. Soft θω-Closure Operator

In this section, we introduce the θω-closure operator that lies strictly between the usual soft closure and soft θ-closure. We provide sufficient conditions for equivalence between the soft θω-closure and usual soft closure operators, and between the soft θω-closure and soft θ-closure operators. We use the soft θω-closure operator to define soft θω-open sets as a new class of soft sets that lies strictly between the class of soft open sets and class of soft θ-open sets. We also demonstrate that the class of soft θω-open sets forms a soft topology. Moreover, we use the soft θω-closure operator and soft θω-open sets to characterize the soft ω-regularity. We also provide a soft product theorem and several soft mapping theorems. Finally, we study the correspondence between the soft topology of the soft θω-open sets of a STS and their generated TSs, and vice versa.

### Definition 2.1 [44]

Let (X, τ,A) be an STS and let FSS(X,A).

(a) A soft point axSP(X,A) is in the soft θ-closure of F (ax ∊̃ Clθ(F)) if Clτ (G) ∩̃ F ≠ 0A for any Gτ with ax ∊̃ G.

(b) F is soft θ-closed if Clθ(F) = F.

(c) F is soft θ-open if 1AF is θ-closed.

(d) The family of all soft θ-open sets in (X, τ,A) is denoted by τθ.

### Theorem 2.2 [44]

Let (X, τ,A) be an STS. Then:

(a) (X, τθ,A) is an STS.

(b) τθτ and τθτ in general.

The following is the main definition of this work.

### Definition 2.3

Let (X, τ,A) be an STS and let FSS(X, A).

(a) A soft point axSP(X,A) is in the soft θω-closure of F (ax ∊̃ Clθω (F),) if Clτω (G) ∩̃ F ≠ 0A for any Gτ with ax ∊̃ G.

(b) F is known as soft θω-closed if Clθω (F) = F.

(c) F is known as soft θω-open if 1AF is θω-closed.

(d) The family of all soft θω-open sets in (X, τ,A) is denoted by τθω.

### Theorem 2.4

Let (X, τ,A) be an STS and let FSS(X, A). Then:

(a) Clτ (F) ⊆̃ Clθω (F) ⊆̃ Clθ(F).

(b) If F is soft θ-closed, F is soft θω-closed.

(c) If F is soft θω-closed, F is soft closed.

Proof

(a) To demonstrate that Clτ (F) ⊆̃ Clθω (F), let ax ∊̃ Clτ (F) and let Gτ such that ax ∊̃ G. As ax ∊̃ Clτ (F), G ∩̃ F ≠ 0A. Because G ⊆̃ Clτω (G), Clτω (G) ∩̃ F ≠ 0A. Thus, ax ∊̃ Clθω (F). To demonstrate that Clθω (F) ⊆̃ Clθ(F), let ax ∊̃ Clθω (F) and let Gτ such that ax ∊̃ G. As ax ∊̃ Clθω (F), Clτω (G) ∩̃ F ≠ 0A. Because Clτω (G) ⊆̃ Clτ (G), Clτ (G) ∩̃ F ≠ 0A. Thus, ax ∊̃ Clθ(F).

(b) Suppose that F is soft θ-closed. Then, Clθ (F) = F, and according to (a), Clθω (F) = F. Hence, F is soft θω-closed.

(c) Suppose that F is soft θω-closed. Then, Clθω (F) = F, and according to (a), Clτ (F) = F. Hence, F is soft closed.

### Theorem 2.5

Let (X, τ,A) be soft ω-locally indiscrete, and let FSS(X,A). Then:

(a) Clτ (F) = Clθω (F).

(b) If F is soft closed, F is soft θω-closed.

Proof

(a) From Theorem 2.4 (a), Clτ (F) ⊆̃ Clθω (F). To show that Clθω (F) ⊆̃ Clτ (F), let ax ∊̃ Clθω (F) and Gτ, with ax ∊̃ G. Then, Clτω (G) ∩̃ F ≠ 0A. Because (X, τ,A) is soft ω-locally indiscrete, Clτω (G) = G; hence, G ∩̃ F ≠ 0A. Therefore, ax ∊̃ Clτ (F).

(b) Suppose that F is soft closed. Then, F = Clτ (F), and according to (a), F = Clθω (F). Hence, F is soft θω-closed.

### Corollary 2.6

Let (X, τ,A) be soft locally indiscrete and let FSS(X,A). Then:

(a) Clτ (F) = Clθω (F).

(b) If F is soft closed, F is soft θω-closed.

Proof

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

### Corollary 2.7

Let (X, τ,A) be soft locally countable and let FSS(X,A). Then:

(a) Clτ (F) = Clθω (F).

(b) If F is soft closed, F is soft θω-closed.

Proof

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

### Theorem 2.8

Let (X, τ,A) be a soft anti-locally countable STS and let FSS(X,A). Then:

(a) Clθ(F) = Clθω (F).

(b) If F is soft θω-closed, F is soft θ-closed.

Proof

(a) Using Theorem 2.4(a), Clθω (F) ⊆̃ Clθ(F). To observe that Clθ(F) ⊆̃ Clθω (F), let ax ∊̃ Clθ(F) and Gτ, with ax ∊̃ G. Then, Clτ (G) ∩̃ F ≠ 0A. Because (X, τ,A) is soft anti-locally countable, according to Theorem 14 of [2], Clτω (G) = Clτ (G); hence, Clτω (G) ∩̃ F ≠ 0A. Therefore, ax ∊̃ Clθω (F).

(b) Suppose that F is soft θω-closed. Then, F = Clθω (F), and according to (a), F = Clθ(F). Hence, F is soft θ-closed.

### Theorem 2.9

For any STS (X, τ,A), τθτθωτ.

Proof

Let Fτθ; then, 1AF is soft θ-closed. Therefore, using Theorem 2.4(b), 1AF is soft θω-closed, and hence, Fτθω. Therefore, τθ. τθω. To demonstrate that τθωτ, let Fτθω; then, 1AF is soft θω-closed. Thus, according to Theorem 2.4(c), 1AF is soft closed. Hence, Fτ.

### Lemma 2.10

Let (X, τ,A) be an STS. Then, for each Fτ, Clθ(F) = Clτ (F).

Proof

Let Fτ. Then, using Theorem 2.4(a), Clτ (F) ⊆̃ Clθ(F). To observe that Clθ(F) ⊆̃ Clτ (F), let ax ∊̃ Clθ(F) and let Gτ, with ax ∊̃ G. Then, Clτ (G) ∩̃ F ≠ 0A. Select by ∊̃ Clτ (G) ∩̃ F. As Fτ, G ∩̃ F ≠ 0A. It follows that Clτ (F) ⊆̃ Clθ(F).

### Theorem 2.11

Let (X, τ, A) be an STS.

(a) If M ⊆̃ N ⊆̃ 1A, Clθω (M) ⊆̃ Clθω (N).

(b) For each M,NSS(X,A), Clθω (M ∪̃ N)= Clθω (M) ∪̃ Clθω (N).

(c) For each MSS(X,A), Clθω (M) is soft closed in (X, τ,A).

(d) For each Fτω, Clθω (F) = Clτ (F).

(e) For each Fτ, Clθ(F) = Clθω (F) = Clτ (F).

Proof

(a) Let ax ∊̃ Clθω (M) and Gτ with ax ∊̃ G. As ax ∊̃ Clθω (M), Clτω (G) ∩̃ M ≠ 0A. Because M ⊆̃ N, we obtain Clτω (G) ∩̃ N ≠ 0A. This demonstrates that ax ∊̃ Clθω (N).

(b) As M ⊆̃ M ∪̃ N and N ⊆̃ M ∪̃ N, according to (a), Clθω (M) ∪̃ Clθω (N) ⊆̃ Clθω (M ∪̃ N). Conversely, let ax ∊̃ 1A−(Clθω (M) ∪̃ Clθω (N)). Then, there exist G,Hτ such that ax ∊̃ G ∩̃ H, Clτω (G) ∩̃ M = 0A, and Clτω (G) ∩̃ N = 0A. Therefore, ax ∊̃ G ∩̃ Hτ and

$Clθω (G∩˜H)∩˜(M∪˜N)=(Clθω (G∩˜H)∩˜M)∪˜(Clθω (G∩˜H)∩˜N)⊆˜(Clθω (G)∩˜M)∪˜(Clθω (H)∩˜N)=0A∪˜0A=0A.$

Hence, ax ∊̃ 1A − (Clθω (M ∪̃ N)).

(c) Let MSS(X,A). We show that 1AClθω (M) ∈ τ. Let ax ∊̃ 1AClθω (M). Then, Gτ exists such that ax ∊̃ G with Clτω (G) ∩̃ M = 0A; thus, G ∩̃ Clθω (M). = 0A. Hence, 1AClθω (M) ∈ τ.

(d) Let Fτω. Then, using Theorem 2.4(a), Clτ (F) ⊆̃ Clθω (F). To observe that Clθω (F) ⊆̃ Clτ (F), we assume to the contrary that there exists ax ∊̃ Clθω (F) ∩̃ (1AClτ (F)). Because ax ∊̃ Clθω(F) and ax ∊̃ (1AClτ (F)) ∈ τ. Then, Clτω (1AClτ (F)) ∩̃ F ≠ 0A. However,

$Clτω (1A-Clτ(F)), ∩˜F⊆˜(1A-Clτ (F))∩˜F⊆˜(1A-F)∩˜F=0A.$

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

### Theorem 2.12

Let (X, τ,A) be an STS. Then:

(a) 0A and 1A are soft θω-closed sets.

(b) The finite soft union of soft θω-closed sets is soft θω-closed.

(c) The arbitrary soft intersection of soft θω-closed sets is soft θω-closed.

Proof

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

(b) We demonstrate that the soft union of two soft θω-closed sets is soft θω-closed. Let M and N be any two soft θω-closed sets. Then, Clθω (M) = M and Clθω (N) = N. Thus, according to Theorem 2.11(b),

$Clθω (M∪˜N)=Clθ,ω(M)∪˜Clθω N)=M∪˜N.$

Hence, M ∪̃ N is soft θω-closed.

(c) Let Fλ be soft θω-closed for all λ ∈ Γ. Then, for all λ ∈ Γ, Fλ = Clθω (Fλ). We show that Clθω ( ∩̃ {Fλ: λ ∈ Γ}) ⊆̃∩̃ {Fλ: λ ∈ Γ}. Let ax ∊̃ Clθω ( ∩̃ {Fλ: λ ∈ Γ}) and let Gτ, with ax ∊̃ G; then, Clτω (G) ∩̃ ( ∩̃ {Fλ: λ ∈ Γ}) ≠ 0A. Thus, for every λ ∈ Γ, Clτω (G) ∩̃ Fλ ≠ 0A. Hence, ax ∊̃ Clθω (Fλ) = Fλ for every λ ∈ Γ. It follows that ax ∊̃∩̃ {Fλ: λ ∈ Γ}.

### Theorem 2.13

For any STS (X, τ,A), (X, τθω,A) is an STS.

Proof

(1) According to Theorem 2.12(a), 0A and 1A are soft θω-closed sets, and thus, 0A, 1Aτθω.

(2) Let M,Nτθω; then, 1AM and 1AN are soft θω-closed sets. Note that

$1A-(M∩˜N)=(1A-M)∪˜(1A-N).$

Then, using Theorem 2.12(b), 1A − (M ∩̃ N) is soft θω-closed. Thus, M ∩̃ Nτθω.

(3) Let Fλτθω for every λ ∈ Γ. Then, {1AFλ: λ ∈ Γ} is a family of soft θω-closed sets. Thus, according to Theorem 2.12(c),

$1A-∪˜ {Fλ:λ∈Γ}=∩˜{1A-Fλ:λ∈Γ}$

is soft θω-closed. Therefore, ∪̃ {Fλ: λ ∈ Γ} ∈ τθω.

### Theorem 2.14

Let (X, τ,A) be an STS and FSS(X,A). Then, Fτθω if and only if, for each ax ∊̃ F, there exists Gτ such that ax ∊̃ G ⊆̃ Clτω (G) ⊆̃ F.

Proof
Necessity

Let Fτθω, and let ax ∊̃ F. Then, we determine that 1AF is soft θω-closed and ax ∊̃ 1A − (1AF). As 1AF is soft θω-closed, Clθω (1AF) = 1AF, and thus, ax ∊̃ 1AClθω (1AF). Hence, there exists Gτ such that ax ∊̃ G and Clτω (G) ∩̃ (1AF) = 0A. Thus, ax ∊̃ G ⊆̃ Clτω (G) ⊆̃ F.

Sufficiency

Suppose that for each ax ∊̃ F, there exists Gτ such that ax ∊̃ G ⊆̃ Clτω (G) ⊆̃ F. Suppose to the contrary that Fτθω. Then, Clθω(1AF) ≠ 1AF, and thus, there exists ax ∊̃ Clθω (1AF) − (1AF). By assumption, there exists Gτ such that ax ∊̃ G ⊆̃ Clτω (G) ⊆̃ F, and thus, ax ∊̃ Gτ with Clτω (G) ∩̃ (1AF) = 0A. This implies that ax ∊̃ 1AClθω (1AF), which is a contradiction.

### Corollary 2.15

Every soft open, soft ω-closed set in an STS is soft θω-open.

Proof

Let (X, τ,A) be an STS, and let F be a soft open and soft ω-closed set. Let ax ∊̃ F. Because F is soft ω-closed, Clτω (F) = F. Select G = F; then, Gτ and ax ∊̃ G = Clτω (G) = F ⊆̃ F. Thus, according to Theorem 2.14, F is soft θω-open.

### Corollary 2.16

Every countable soft open set in an STS is soft θω-open.

Proof

Using Theorem 2(d) of [2], countable soft sets in an STS are soft ω-closed. Therefore, using Corollary 2.15, we obtain the result.

### Theorem 2.17

For any STS (X, τ,A), the following are equivalent:

(a) (X, τ,A) is soft ω-regular.

(b) τ = τθω.

(c) For every FSS(X,A), Clθω (F) = Clτ (F).

Proof

(a) ⇒ (b): Suppose that (X, τ,A) is soft ω-regular. To observe that ττθω, let Fτ and let ax ∊̃ F. Because (X, τ,A) is soft ω-regular, according to Theorem 13 of [6], there exists Gτ such that ax ∊̃ G ⊆̃ Clτω(G) ⊆̃ F. Therefore, using Theorem 2.14, we obtain Fτθω. However, from Theorem 2.9, we obtain τθωτ.

(b) ⇒ (c): Suppose that τ = τθω and let FSS(X,A). To verify that Clθω (F), ⊆̃ Clτ (F), let ax ∊̃ 1AClτ (F). Because 1AClτ (F) ∈ τ, using (b), 1AClτ (F) ∈ τθω. Thus, according to Theorem 2.14, there exists Gτ such that ax ∊̃ G ⊆̃ Clτω (G) ⊆̃ 1AClτ (F). Therefore, we obtain ax ∊̃ Gτ and Clτω (G) ∩̃ F ⊆̃ Clτω (G) ∩̃ Clτ (F) = 0A. Hence, ax ∊̃ 1AClθω (F). However, using Theorem 2.4(a), we obtain Clτ (F) ⊆̃ Clθω (F).

(c) ⇒ (a): Suppose that Clθω (F) = Clτ (F) for every FSS(X,A). Let Fτ and let ax ∊̃ F. Then, 1AF is soft closed and using (c), Clθω (1AF) = Clτ (1AF) = 1AF. Therefore, ax ∊̃ 1AClθω (1AF), and thus, there exists Gτ such that ax ∊̃ G and Clτω (G) ∩̃ (1AF) = 0A. Hence, ax ∊̃ G ⊆̃ Clτω (G) ⊆̃ F. Therefore, according to Theorem 13 of [6], (X, τ,A) is soft ω-regular.

### Remark

The inclusions in Theorem 2.9 are not equalities in general.

### Example 2.18

Let X = ℝ, A = {a, b}, and for all eA}.

Then,

$τθω={F∈SS(X,A):F(e)∈{∅, ℝ, ℕ} for all e∈A},$

and

$τθ={F∈SS(X,A):F(e)∈{∅, ℝ} for all e∈A}.$

### Lemma 2.19 [5]

Let (X, τ,A) be an STS and FSS(X, A). Then, for all aA, Clτa (F (a)) ⊆ (Clτ (F)) (a).

### Theorem 2.20

Let (X, τ,A) be an STS. If Fτθ, F(a) ∈ (τa)θ for all aA.

Proof

Suppose that Fτθ and let aA. Let xF(a) exists Gτ such that ax ∊̃ G ⊆̃ Clτ (G) ⊆̃ F. Thus, we obtain G(a) ∈ τa, and using Lemma 2.19, xG(a) ⊆ Clτa (G(a)) ⊆ (Clτ (G)) (a) ⊆ F(a). Therefore, F(a) ∈ (τa)θ.

### Theorem 2.21

Let X be an initial universe and let A be a set of parameters. Let {ℑa: aA} be an indexed family of topologies on X. Then, $F∈(⊕a∈Aℑa)θ$ if and only if F(a) ∈ (ℑa)θ for all aA.

Proof
Necessity

Suppose that $F∈(⊕a∈Aℑa)θ$. Then, according to Theorem 2.20, $F(a)∈((⊕a∈Aℑa)a)θ$

for all aA. However, from Theorem 3.7 of [1], $(⊕a∈Aℑa)a=ℑa$ for all aA. This ends the proof.

Sufficiency

Suppose that F(a) ∈ (ℑa)θ for all aA. Let ax ∊̃ F; then, xF(a) ∈ (ℑa)θ. Therefore, there exists V ∈ ℑa such that xVCla (V ) ⊆ F(a). Thus, we obtain ax ∊̃ aV ⊆̃ aCla (V ) ⊆̃ F, with $aV⊕a∈Aℑa$. Furthermore, according to Proposition 2 of [6], $aClℑa(V)=Cl⊕a∈Aℑa(aV)$. Hence, $F∈(⊕a∈Aℑa)θ$.

### Corollary 2.22

Let (X, ℑ) be a TS and let A be a set of parameters. Then, F ∈ (τ (ℑ))θ if and only if F(a) ∈ ℑθ for all aA.

Proof

For each aA, set ℑa = ℑ. Then, $τ (ℑ)=⊕a∈Aℑa$. Thus, using Theorem 2.21, we obtain the result.

### Theorem 2.23

Let (X, τ,A) be an STS. If Fτθω, F(a) ∈ (τa)θω for all aA.

Proof

Suppose that Fτθω and let aA. Let xF(a); then, ax ∊̃ F. As Fτθω, there exists Gτ such that ax ∊̃ G ⊆̃ Clτω (G) ⊆̃ F. Thus, we obtain G(a) ∈ τa, and according to Lemma 2.19 and Theorem 7 of [2], xG(a) ⊆ Cl(τa)ω, (G(a)) = Cl(τω)a (G(a)) ⊆ (Clτω (G)) (a) ⊆ F(a). Therefore, F(a) ∈ (τa)θω.

### Theorem 2.24

Let X be an initial universe and let A be a set of parameters. Let {ℑa: aA} be an indexed family of topologies on X. Then, $F∈(⊕a∈Aℑa)θω$ if and only if F(a) ∈ (ℑa)θω for all aA.

Proof
Necessity

Suppose that $F∈(⊕a∈Aℑa)θω$. Then, using Theorem 2.23, $F(a)∈((⊕a∈Aℑa)a)θω$ for all aA. However, according to Theorem 3.7 of [1], $(⊕a∈Aℑa)a=ℑa$ for all aA. This ends the proof.

Sufficiency

Suppose that F(a) ∈ (ℑa)θω for all aA. Let ax ∊̃ F; then, xF(a) ∈ (ℑa)θω. Thus, there exists V ∈ ℑa such that xVCl(ℑa)ω (V ) ⊆ F(a). Thus, we have ax ∊̃ aV ⊆̃ aCl(a)ω(V) ⊆̃ F with $aV∈⊕a∈Aℑa$. Moreover, according to Proposition 2 in [6], $aCl(ℑa)ω(V)=Cl⊕a∈A(ℑa)ω(aV)$. Moreover, using Theorem 8 of [2], $⊕a∈A(ℑa)ω=(⊕a∈Aℑa)ω$. Hence, $F∈(⊕a∈Aℑa)θω$.

### Corollary 2.25

Let (X, ℑ) be a TS and let A be a set of parameters. Then, F ∈ (τ (ℑ))θω if and only if F(a) ∈ ℑθω for all aA.

Proof

For each aA, set ℑa = ℑ. Then, $τ (ℑ)=⊕a∈Aℑa$. Thus, using Theorem 2.24, we obtain the result.

### Theorem 2.26

Let (X, τ, A) and (Y, σ, B) be two STSs. Let FSS(X,A) and GSS(Y,B). If F ×G ∈ (τ * σ)θω, Fτθω and Gσθω.

Proof

Let ax ∊̃ F and by ∊̃ G, then (a, b)(x,y) ∊̃ F × G. Because F × G ∈ (τ * σ)θω, there exists Mτ * σ such that (a, b)(x,y) ∊̃ M ⊆̃ Cl(τ*σ)ω (M) ⊆̃ F × G. Select Kτ and Nσ such that (a, b)(x,y) ∊̃ K × N ⊆̃ M. Then, according to Proposition 3(b) of [6], (a, b)(x,y) ∊̃ K × N ⊆̃ Clτω (K) × Clσω (N) ⊆̃ Cl(τ*σ)ω (K × N) ⊆̃ Cl(τ*σ)ω (M) ⊆̃ F × G.

Thus, we obtain ax ∊̃ K ⊆̃ Clτω (K) ⊆̃ F and by ∊̃ N ⊆̃ Clτω (N) ⊆̃ G. Therefore, Fτθω and Gσθω.

### Question 2.27

Let (X, τ,A) and (Y, σ,B) be two STSs. Let Fτθω and Gσθω. Is it true that F × G ∈ (τ * σ)θω ?

If fpu: (X, τ,A) → (Y, σ,B) is a soft closed function, it is clear that fpu: (X, τ,A) → (Y, σω,B) is soft closed. The following example shows that the converse is not true in general.

### Example 2.28

Let τ = {FSS(ℝ, {a}): F(a) = ∅︀ or ℝ − F(a) is finite}.

Define u: {a} → {a} and p: ℝ → ℝ using u(a) = a and

$p(x)={x,if x∈ℤ,2,if x∈ℝ-ℤ.$

For every soft closed set F of (ℝ, τ, {a}), (fpu(F)) (a) ⊆ ℤ. Thus, fpu: (X, τ, {a}) → (X, τω, {a}) is soft closed. Because 1A is soft closed in (X, τ, {a}) but fpu (1A) is not soft closed in (X, τ, {a}), fpu: (X, τ, {a}) → (X, τ, {a}) is not soft closed.

### Theorem 2.29

Let fpu: (X, τ,A) → (Y, σ,B) be a soft function. If fpu: (X, τ,A) → (Y, σ,B) is soft open and fpu: (X, τ,A) → (Y, σω,B) is soft closed, fpu: (X, τθ,A) → (Y, σθω,B) is soft open.

Proof

Let Gτθ and let by ∊̃ fpu (G). Select ax ∊̃ G such that by = fpu (ax). As ax ∊̃ Gτθ, there exists Mτ such that ax ∊̃ M ⊆̃ Clτ (M) ⊆̃ G. Therefore, by = fpu (ax) ∊̃ fpu(M) ⊆̃ fpu (Clτ (M)) ⊆̃ fpu (G). Because fpu: (X, τ,A) → (Y, σ,B) is soft open, fpu(M) ∈ σ. As fpu: (X, τ,A) → (Y, σω,B) is soft closed, fpu (Clτ (M)) is soft ω-closed in (Y, σ,B), and thus, Clτω (fpu(M)) ⊆̃ fpu (Clτ (M)). Therefore, fpu (G) ∈ σθω. It follows that fpu: (X, τθ,A) → (Y, σθω,B) is soft open.

### Theorem 2.30

Let fpu: (X, τ,A) → (Y, σ,B) be a soft function. If fpu: (X, τ,A) → (Y, σ,B) is soft open and fpu: (X, τω,A) → (Y, σω,B) is soft closed, fpu: (X, τθω,A) → (Y, σθω,B) is soft open.

Proof

Let Gτθω and let by ∊̃ fpu (G). Select ax ∊̃ G such that by = fpu (ax). As ax ∊̃ Gτθω, there exists Mτ such that ax ∊̃ M ⊆̃ Clτω (M) ⊆̃ G. Thus, by = fpu (ax) ∊̃ fpu(M) ⊆̃ fpu (Clτω (M)) ⊆̃ fpu (G). Because fpu: (X, τ,A) → (Y, σ, B) is soft open, f(M) ∈ σ. As fpu: (X, τω,A) → (Y, σω,B) is soft closed, fpu (Clτω (M)) is ω-closed in (Y, σ,B); therefore, Clτω (fpu(M)) ⊆̃ fpu(Clτω (M)) ⊆̃ fpu (G). It follows that fpu (G) ∈ σθω.

### Theorem 2.31

Let fpu: (X, τ,A) → (Y, σ,B) be a soft function. If both fpu: (X, τ,A) → (Y, σ,B) and fpu: (X, τω,A) → (Y, σω,B) are soft continuous, fpu: (X, τθω,A) → (Y, σθω,B) is soft continuous.

Proof

Let Hσθω and let $ax∈˜fpu-1(H)$. Then, fpu (ax) ∊̃ H; thus, we determine Mσ such that fpu (ax) ∊̃ M ⊆̃ Clτω (M) ⊆̃ H. Therefore, $ax∈˜fpu-1(M)⊆˜fpu-1(Clτω(M))⊆˜fpu-1(H)$. As fpu: (X, τ,A) → (Y, σ,B) is soft continuous, $fpu-1(M)∈τ$. Because fpu: (X, τω,A) → (Y, σω,B) is soft continuous, and $fpu-1(Clτω(M))$ is soft ω-closed in (X, τ,A); thus, $Clτω (fpu-1(M))⊆˜fpu-1(Clτω(M))⊆˜fpu-1(H)$. Hence, $fpu-1(H)∈τθω$. It follows that fpu: (X, τθω,A) → (Y, σθω,B) is soft continuous.

### 3. Soft θω-Continuity

In this section, we introduce and investigate soft θω-continuity as a strong form of soft θ-continuity.

### Definition 3.1 [44]

A soft function fpu: (X, τ,A) → (Y, σ,B) is said to be soft θ-continuous if for every axSP(X,A) and every Gσ such that fpu(ax) ∊̃ G, there exists Hτ such that ax ∊̃ H and fpu(Clτ (H)) ⊆̃ Clσ(G).

### Theorem 3.2

Let p: (X, ℑ) → (Y, ℵ) be a function between two TSs and let u: AB be a function between two sets of parameters. Then, fpu: (X, τ(ℑ),A) → (Y, τ(ℵ),B) is soft θ-continuous if and only if p: (X, ℑ) → (Y, ℵ) is θ-continuous.

Proof
Necessity

Suppose that fpu: (X, τ(ℑ),A) → (Y, τ(ℵ),B) is soft θ-continuous. Let xX and V ∈ ℵ such that p(x) ∈ V. Select aA; then, fpu(ax) = u(a)p(x) ∊̃ u(a)V. As fpu: (X, τ(ℑ),A) → (Y, τ(ℵ),B) is soft θ-continuous, there exists Hτ such that ax ∊̃ H and fpu(Clτ()(H)) ⊆̃ Clτ(ℵ)(u(a)V ). Thus, p((Clτ()(H))(a)⊆(fpu(Clτ()(H))) (u(a)) ⊆ (Clτ(ℵ)(u(a)V ))(u(a)). According to Lemma 2.19, p(Cl(H(a))) = p(Cl(τ())a (H(a))) ⊆ p((Clτ()(H))(a)). However, using Proposition 2 of [6], (Clτ(ℵ)(u(a)V )) (u(a)) = u(a)Cl(V )(u(a)) = Cl(V ). Therefore, we obtain xH(a) ∈ ℑ and p(Cl(H(a))) ⊆ Cl(V ). Hence, p: (X, ℑ) → (Y, ℵ) is θ-continuous.

### Sufficiency

Suppose that p: (X, ℑ) → (Y, ℵ) is θ-continuous. Let axSP(X,A) and let Gτ (ℵ) such that fpu (ax) = u(a)p(x) ∊̃ G. Then, p(x) ∈ G(u(a)) ∈ ℵ. Because p: (X, ℑ) → (Y, ℵ) is θ-continuous, there exists S ∈ ℑ such that x ∊̃ S and p (Cl(S)) ⊆ Cl(G(u(a))). According to Lemma 2.19, Cl(G(u(a))) = Cl(τ(ℵ))u (a) (G(u(a))) ⊆ (Cl(G)) (u(a)). Furthermore, from Proposition 2 of [6], Clτ() (aS) = aCl (S), and thus, fpu(Clτ()(aS)) = fpu(aCl (S)) = u(a)p(Cl (S)). Therefore, we obtain ax ∊̃ aSτ (ℑ) and fpu (Clτ()(aS)) ⊆̃ Clτ(ℵ)(G). Hence, fpu: (X, τ( ℑ),A) → (Y, τ(ℵ),B) is soft θ-continuous.

### Definition 3.3

A soft function fpu: (X, τ,A) → (Y, σ,B) is said to be soft θω-continuous if for every axSP(X,A) and every Gσ such that fpu(ax) ∊̃ G, there exists Hτ such that ax ∊̃ H and fpu(Clτ (H)) ⊆̃ Clσω (G).

### Theorem 3.4

Let p: (X, ℑ) → (Y, ℵ) be a function between two TSs and let u: AB be a function between two sets of parameters. Then, fpu: (X, τ(ℑ),A) → (Y, τ(ℵ),B) is soft θω-continuous if and only if p: (X, ℑ) → (Y, ℵ), is θω-continuous.

Proof
Necessity

Suppose that fpu: (X, τ(ℑ),A) → (Y, τ(ℵ),B) is soft θω-continuous. Let xX and V ∈ ℵ such that p(x) ∈ V. Select aA. Then, fpu(ax) = u(a)p(x) ∊̃ u(a)V. As fpu: (X, τ(ℑ),A) → (Y, τ(ℵ),B) is soft θω-continuous, there exists Hτ such that ax ∊̃ H and fpu(Clτ()(H)) ⊆̃ Cl(τ(ℵ))ω (u(a)V ). Thus, p((Clτ()(H))(a)) ⊆ (fpu(Clτ()(H)))(u(a)) ⊆ (Clτ(ℵ)(u(a)V ))(u(a)). According to Lemma 2.19, p(Cl(H(a))) = p(Cl(τ())a (H(a))) ⊆ p((Clτ()(H))(a)). In contrast, using Proposition 2 of [6], (Cl(τ(ℵ))ω (u(a)V ))(u(a)) = u(a)Clω (V )(u(a)) = Clω (V ). Therefore, we obtain xH(a) ∈ ℑ and p(Cl(H(a))),⊆ Clω (V ). Hence, p: (X, ℑ) → (Y, ℵ) is θω-continuous.

### Sufficiency

Suppose that p: (X, ℑ) → (Y, ℵ) is θω-continuous. Let axSP(X,A) and let Gτ (ℵ) such that fpu (ax) = u (a)p(x) ∊̃ G. Then, p(x) ∈ G(u(a)) ∈ ℵ. Because p: (X, ℑ) → (Y, ℵ) is θω-continuous, there exists S ∈ ℑ such that x ∊̃ S and p(Cl(S)) ⊆ Clω (G(u(a))). According to Lemma 2.19, Clω (G(u(a))) = Cl(τ(ℵω))u(a) (G(u(a))) ⊆ (Clω (G)) (u(a)). Furthermore, from Proposition 2 of [6], Clτ()(aS) = aCl (S), and thus, fpu(Clτ()(aS)) = fpu(aCl (S)) = u(a)p(Cl (S)). Therefore, we obtain ax ∊̃ aSτ (ℑ) and fpu(Clτ()(aS)) ⊆̃ Cl(τ(ℵ))ω (G). Hence, fpu: (X, τ (ℑ),A) → (Y, τ (ℵ),B) is soft θω-continuous.

### Theorem 3.5

Every soft θω-continuous function is soft θ-continuous.

Proof

Let fpu: (X, τ,A) → (Y, σ,B) be soft θω-continuous. Let axSP(X,A). and Gσ such that fpu(ax) ∊̃ G. Based on the soft θω-continuity of fpu, there exists Hτ such that ax ∊̃ H and fpu(Clτ (H)) ⊆̃ Clσω (G). Because Clσω (G) ⊆̃ Clσ(G), fpu(Clτ (H)) ⊆̃ Clσ(G). Therefore, fpu is soft θ-continuous.

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

### Example 3.6

Let X = Y = ℕ, ℑ = {∅︀, X}, ℵ = {∅︀}, ∪{UY: YU is finite }, A = ℝ, and B = {b, c}. Consider the functions p: XY and u: AB, which are defined as p(x) = x for all xX, u(t) = b if and u(t) = c if . Then, as demonstrated in Example 2.4 of [40], p: (X, ℑ) → (Y, ℵ) is θ-continuous but not θω-continuous. Hence, according to Theorems 3.2 and 3.4, fpu: (X, τ(ℑ),A) → (Y, τ(ℵ),B) is soft θ-continuous but not soft θω-continuous.

### Theorem 3.7

If fpu: (X, τ,A) → (Y, σ,B) is a soft θ-continuous function with (Y, σ,B) in which is soft anti-locally countable, fpu: (X, τ,A) → (Y, σ,B) is soft θω-continuous.

Proof

Suppose that fpu: (X, τ,A) → (Y, σ,B) is a soft θ-continuous function. Let axSP(X,A) and let Gσ such that fpu(ax) ∊̃ G. Then, according to the soft θ-continuity of fpu, there exists Hτ such that ax ∊̃ H and fpu(Clτ (H)) ⊆̃ Clσ(G). As (Y, σ,B) is soft anti-locally countable, according to Theorem 14 of [2], Clσ(G) = Clσω (G), and thus, fpu(Clτ (H)) ⊆̃ Clσω (G). Therefore, fpu is soft θω-continuous.

### Theorem 3.8 [51]

Every soft continuous function is soft θ-continuous, but the converse is not true.

The following two examples demonstrate that soft continuity and soft θω-continuity are independent concepts:

### Example 3.9

Let X = Y = ℝ, ℑ = {∅︀, X}, ℵ = {∅︀}, ∪{UY: YU is finite }, A = ℝ, and B = {b, c}. Consider the functions p: XY and u: AB, which are defined as p(x) = x for all xX, u(t) = b if and u(t) = c if . Then, as proven in Example 2.7 of [40], p: (X, ℑ) → (Y, ℵ) is θω-continuous, but not continuous. Hence, according to Theorem 3.4 and Theorem 5.31 of [1], fpu: (X, τ(ℑ),, A) → (Y, τ(ℵ), B) is soft θω-continuous but not soft continuous.

### Example 3.10

Let X = Y = ℕ, ℑ = ℵ = {∅︀, X, {1}}, A = ℝ, and B = {b, c}. Consider the functions p: XY and u: AB, which are defined as p(x) = x for all xX, u(t) = b if and u(t) = c if . Then, as shown in Example 2.8 of [40], p: (X, ℑ) → (Y, ℵ) is continuous, but not θω-continuous. Hence, according to Theorem 5.31 of [1] and Theorem 3.4, fpu: (X, τ(ℑ),A) → (Y, τ(ℵ),B) is soft continuous, but not soft θω-continuous.

The following result provides a sufficient condition for a soft θω-continuous function to be soft continuous:

### Theorem 3.11

If fpu: (X, τ,A) → (Y, σ,B) is a soft θω-continuous function in which (Y, σ,B) is soft ω-regular, fpu: (X, τ,A) → (Y, σ,B) is soft continuous.

Proof

Suppose that fpu: (X, τ,A) → (Y, σ,B) is a soft θω-continuous function. Let axSP(X,A) and let Gσ such that fpu(ax) ∊̃ G. As (Y, σ,B) is soft ω-regular, there exists Hσ such that fpu(ax) ∊̃ H ⊆̃ Clσω (H) ⊆̃ G. Because fpu is soft θω-continuous, there exists Mτ such that ax ∊̃ M and fpu(Clτ (M)) ⊆̃ Clσω (H). Therefore, we obtain

$f p u ( M ) ⊆ ˜ f p u ( C l τ ( M ) ) ⊆ ˜ C l σ ω ( H ) ⊆ ˜ G$

Hence, fpu is soft continuous.

### Theorem 3.12

Let fpu: (X, τ,A) → (Y, σ,B) be a soft function such that for every axSP(X,A) and every Gσ such that fpu(ax) ∊̃ G, there exists Hτθ such that ax ∊̃ H and fpu(H) ⊆̃ Clσω(G); then, fpu is soft θω-continuous.

Proof

Let axSP(X,A) and let Gσ such that fpu(ax) ∊̃ G. Then, by assumption, there exists Hτθ such that ax ∊̃ H and fpu(H) ⊆̃ Clσω (G). Because ax ∊̃ Hτθ, there exists Mτ such that ax ∊̃ M ⊆̃ Clτ (M) ⊆̃ H. Therefore, we obtain ax ∊̃ Mτ with fpu(Clτ (M)) ⊆̃ fpu(H) ⊆̃ Clσω(G). Hence, fpu: (X, τ,A) →.(Y, σ,B) is soft θω-continuous.

### Theorem 3.13

Let fpu: (X, τ,A) → (Y, σ,B) be a soft function such that for every Gσ, $f p u - 1 ( C l σ ω ( G ) ) ∈ τ θ$, fpu is soft θω-continuous.

Proof

Using Theorem 3.12, it is sufficient to prove that for every axSP(X,A) and every Gσ such that fpu(ax) ∊̃ G, there exists Hτθ such that ax ∊̃ H and fpu(H) ⊆̃ Clσω (G). Let axSP(X,A) and let Gσ such that fpu(ax) ∊̃ G. Set $H = f p u - 1 ( C l σ ω ( G ) )$. Then, by assumption, Hτθ. Furthermore, because fpu(ax) ∊̃ G, $a x ∈ ˜ f p u - 1 ( G ) ⊆ ˜ f p u - 1 ( C l σ ω ( G ) ) = H$. Moreover, as $H = f p u - 1 ( C l σ ω ( G ) ) , f p u ( H ) = f p u ( f p u - 1 ( C l σ ω ( G ) ) ) ⊆ ˜ C l σ ω ( G )$.

### Theorem 3.14

If fpu: (X, τ,A) → (Y, σ,B) is soft θω-continuous, for every FSS(X,A), fpu(Clθ(F)) ⊆̃ Clθω (fpu (F)).

Proof

Suppose that fpu: (X, τ,A) → (Y, σ,B) is soft θω-continuous and let FSS(X,A). Let by ∊̃ fpu (Clθ(F)). To observe that by ∊̃ Clθω (fpu(F)), let Gσ such that by ∊̃ G. Because by ∊̃ fpu (Clθ(F)), there exists ax ∊̃ Clθ(F) such that by = fpu (ax). Because fpu is soft θω-continuous, there exists Hτ such that ax ∊̃ H and fpu(Clτ (H)) ⊆̃ Clσω (G). As ax ∊̃ Hτ and ax ∊̃ Clθ(F), Clτ (H) ∩̃ F ≠ 0A, and thus, fpu(Clτ (H) ∩̃ F) ≠ fpu (0A) = 0B. Because fpu(Clτ (H) ∩̃ F) ⊆̃ fpu(Clτ(H)) ∩̃ fpu(F) ⊆̃ Clσω (G) ∩̃ fpu(F), Clσω(G) ∩̃ fpu(F) ≠ 0B. It follows that by ∊̃ Clθω (fpu(F)).

### 4. Conclusion

We have defined and investigated the θω-closure operator. Furthermore, we defined soft θω-open sets as a new class of soft sets that form a new soft topology. We have also introduced the soft θω-continuity as a strong form of θ-continuity. We have presented several characterizations, implications, and examples relating to these concepts. The following topics could be considered in future studies: solving an open question raised in this work, and defining soft separation axioms via soft θω-open sets.

### References

1. Al Ghour, S, and Bin-Saadon, A (2019). On some generated soft topological spaces and soft homogeneity. Heliyon. 5. article no. e02061
2. Al Ghour, S, and Hamed, W (2020). On two classes of soft sets in soft topological spaces. Symmetry. 12. article no. 265
3. Al Ghour, S, and Irshedat, B (2017). The topology of θω-open sets. Filomat. 31, 5369-5377. https://doi.org/10.2298/FIL1716369A
4. Molodtsov, D (1999). Soft set theory—first results. Computers & Mathematics with Applications. 37, 19-31. https://doi.org/10.1016/S0898-1221(99)00056-5
5. Shabir, M, and Naz, M (2011). On soft topological spaces. Computers & Mathematics with Applications. 61, 1786-1799. https://doi.org/10.1016/j.camwa.2011.02.006
6. Al Ghour, S (2021). Weaker forms of soft regular and soft T2 soft topological spaces. Mathematics. 9. article no. 2153
7. Al Ghour, S (2021). Soft minimal soft sets and soft prehomogeneity in soft topological spaces. International Journal of Fuzzy Logic and Intelligent Systems. 21, 269-279. https://doi.org/10.5391/IJFIS.2021.21.3.269
8. Kocinac, LD, Al-shami, TM, and Cetkin, V (2021). Selection principles in the context of soft sets: Menger spaces. Soft Computing. 25, 12693-12702. https://doi.org/10.1007/s00500-021-06069-6
9. Al-shami, TM (2021). New soft structure: infra soft topological spaces. Mathematical Problems in Engineering. 2021. article no. 3361604
10. Al-shami, TM (2021). Infra soft compact spaces and application to fixed point theorem. Journal of Function Spaces. 2021. article no. 3417096
11. Asaad, BA, Al-shami, TM, and Mhemdi, A (2021). Bioperators on soft topological spaces. AIMS Mathematics. 6, 12471-12490. https://doi.org/10.3934/math.2021720
12. Al-shami, TM, and Mhemdi, A (2021). Belong and nonbelong relations on double-framed soft sets and their applications. Journal of Mathematics. 2021. article no. 9940301
13. Alkhazaleh, S, and Marei, EA (2021). New soft rough set approximations. International Journal of Fuzzy Logic and Intelligent Systems. 21, 123-134. https://doi.org/10.5391/IJFIS.2021.21.2.123
14. Al Ghour, S (2021). Strong form of soft semi-open sets in soft topological spaces. International Journal of Fuzzy Logic and Intelligent Systems. 21, 159-168. https://doi.org/10.5391/IJFIS.2021.21.2.159
15. Al Ghour, S (2021). Soft ω-paracompactness in soft topological spaces. International Journal of Fuzzy Logic and Intelligent Systems. 21, 57-65. https://doi.org/10.5391/IJFIS.2021.21.1.57
16. Al-shami, TM, and Abo-Tabl, ESA (2021). Connectedness and local connectedness on infra soft topological spaces. Mathematics. 9. article no. 1759
17. Mousarezaei, R, and Davvaz, B (2021). On soft topological polygroups and their examples. International Journal of Fuzzy Logic and Intelligent Systems. 21, 29-37. https://doi.org/10.5391/IJFIS.2021.21.1.29
18. Oztunc, S, Aslan, S, and Dutta, H (2021). Categorical structures of soft groups. Soft Computing. 25, 3059-3064. https://doi.org/10.1007/s00500-020-05362-0
19. Al-Shami, TM, and Abo-Tabl, EA (2021). Soft α-separation axioms and α-fixed soft points. AIMS Mathematics. 6, 5675-5694. https://doi.org/10.3934/math.2021335
20. Al-Shami, TM (2021). Bipolar soft sets: relations between them and ordinary points and their applications. Complexity. 2021. article no. 6621854
21. Al-Shami, TM, Alshammari, I, and Asaad, BA (2020). Soft maps via soft somewhere dense sets. Filomat. 34, 3429-3440. https://doi.org/10.2298/FIL2010429A
22. Oguz, G (2020). Soft topological transformation groups. Mathematics. 8. article no. 1545
23. Min, WK (2020). On soft ω-structures defined by soft sets. International Journal of Fuzzy Logic and Intelligent Systems. 20, 119-123. https://doi.org/10.5391/IJFIS.2020.20.2.119
24. Cetkin, V, Guner, E, and Aygun, H (2020). On 2S-metric spaces. Soft Computing. 24, 12731-12742. https://doi.org/10.1007/s00500-020-05134-w
25. El-Shafei, ME, and Al-Shami, TM (2020). Applications of partial belong and total non-belong relations on soft separation axioms and decision-making problem. Computational and Applied Mathematics. 39. article no. 138
26. Alcantud, JCR (2020). Soft open bases and a novel construction of soft topologies from bases for topologies. Mathematics. 8. article no. 672
27. Bahredar, AA, and Kouhestani, N (2020). On ∊-soft topological semigroups. Soft Computing. 24, 7035-7046. https://doi.org/10.1007/s00500-020-04826-7
28. Al-shami, TM, and El-Shafei, ME (2020). Partial belong relation on soft separation axioms and decision-making problem, two birds with one stone. Soft Computing. 24, 5377-5387. https://doi.org/10.1007/s00500-019-04295-7
29. Al-shami, TM, Kocinac, LD, and Asaad, BA (2020). Sum of soft topological spaces. Mathematics. 8. article no. 990
30. Al-Saadi, HS, and Min, WK (2017). On soft generalized closed sets in a soft topological space with a soft weak structure. International Journal of Fuzzy Logic and Intelligent Systems. 17, 323-328. https://doi.org/10.5391/IJFIS.2017.17.4.323
31. Velichko, NV (1966). H-closed topological spaces. Mat Sb. 70, 98-112.
32. Hassan, JA, and Labendia, MA (2022). θs-open sets and θscontinuity of maps in the product space. Journal of Mathematics and Computer Science. 25, 182-190. https://doi.org/10.22436/jmcs.025.02.07
33. Osipov, AV (2021). On the cardinality of S(n)-spaces. Quaestiones Mathematicae. 44, 121-128. https://doi.org/10.2989/16073606.2019.1672112
34. Periyasamy, P, and Ramesh, PR (2020). Local δ-closure functions in ideal topological spaces. Advances in Mathematics: Scientific Journal. 9, 2427-2436. https://doi.org/10.37418/amsj.9.5.6
35. Ramirez-Paramo, A (2019). A generalization of some cardinal function inequalities. Topology and its Applications. 256, 228-234. https://doi.org/10.1016/j.topol.2019.02.009
36. Babinkostova, L, Pansera, BA, and Scheepers, M (2019). Selective versions of θ-density. Topology and its Applications. 258, 268-281. https://doi.org/10.1016/j.topol.2019.02.061
37. Modak, S, and Noiri, T (2019). Some generalizations of locally closed sets. Iranian Journal of Mathematical Sciences and Informatics. 14, 159-165. https://doi.org/10.7508/ijmsi.2019.01.014
38. Reyes, JD, and Morales, AR (2018). The weak Urysohn number and upper bounds for cardinality of Hausdorff spaces. Houston Journal of Mathematics. 44, 1389-1398.
39. Al Ghour, S, and Al-Zoubi, S (2021). A new class between theta open sets and theta omega open sets. Heliyon. 7. article no. e05996
40. Al Ghour, S, and Irshidat, B (2020). On θω continuity. Heliyon. 6. article no. e03349
41. Butanas, LLL, and Labendia, MA (2020). θω-connected space and θω-continuity in the product space. Poincare Journal of Analysis and Applications. 7, 79-88. https://doi.org/10.46753/pjaa.2020.v07i01.008
42. Latif, RM (2020). Theta-ω-mappings in topological spaces. WSEAS Transactions on Mathematics. 19, 186-207. https://doi.org/10.37394/23206.2020.19.18
43. Al Ghour, S, and El-Issa, S (2019). θω-Connectedness and ω-R1 properties. Proyecciones (Journal of Mathematics). 38, 921-942. https://doi.org/10.22199/issn.0717-6279-2019-05-0059
44. Georgiou, DN, Megaritis, AC, and Petropoulos, VI (2013). On soft topological spaces. Applied Mathematics & Information Sciences. 7, 1889-1901. https://doi.org/10.12785/amis/070527
45. Al-shami, TM, and Kocinac, LD (2019). The equivalence between the enriched and extended soft topologies. Applied and Computational Mathematics. 18, 149-162.
46. Das, S, and Samanta, S (2013). Soft metric. Annals of Fuzzy Mathematics and Informatics. 6, 77-94.
47. Nazmul, S, and Samanta, SK (2013). Neighbourhood properties of soft topological spaces. Annals of Fuzzy Mathematics and Informatics. 6, 1-15.
48. Babitha, KV, and Sunil, J (2010). Soft set relations and functions. Computers & Mathematics with Applications. 60, 1840-1849. https://doi.org/10.1016/j.camwa.2010.07.014
49. Aygunoglu, A, and Aygun, H (2012). Some notes on soft topological spaces. Neural Computing and Applications. 21, 113-119. https://doi.org/10.1007/s00521-011-0722-3
50. Sayed, ME, and El-Bably, MK (2017). Soft simply open sets in soft topological space. Journal of Computational and Theoretical Nanoscience. 14, 4100-4103. https://doi.org/10.1166/jctn.2017.6792
51. Fomin, S (1943). Extensions of topological spaces. Annals of Mathematics. 43, 471-480. https://doi.org/10.2307/1968976
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 42, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9339752793312073, "perplexity": 15920.02503971437}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337906.7/warc/CC-MAIN-20221007014029-20221007044029-00590.warc.gz"}