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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.

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.

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.

This is a contradiction.

(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.

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)).

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.

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

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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

Copyright © 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.

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 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.

This is a contradiction.

(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.

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