Soft -Open Sets and Soft -Continuity

Samer Al Ghour

doi:10.5391/IJFIS.2022.22.1.89

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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

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

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Abstract
1. Introduction
2. Soft θ_ω-Closure Operator
3. Soft θ_ω-Continuity
4. Conclusion
Conflict of Interest
References
Biography

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

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

1. Introduction

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Abstract
1. Introduction
2. Soft θ_ω-Closure Operator
3. Soft θ_ω-Continuity
4. Conclusion
Conflict of Interest
References
Biography

This paper follows the concepts and terminology that appear in [1–3]. In this paper, TS and STS denote the topological space and soft topological space, respectively. The concept of soft sets, which was introduced by Molodtsov [4] in 1999, is a general mathematical tool for dealing with uncertainty. Let 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 0_A and 1_A, respectively. As a contemporary structure of mathematics, STSs were defined in [5] as follows. An STS is a triplet (X, τ, A), where τ ⊆ SS (X, A), τ contains 0_A and 1_A, 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 F ∈ SS(X, A); then, F is said to be a soft open set in (X, τ, A) if F ∈ τ, and F is said to be a soft closed set in (X, τ, A) if 1_A − F is a soft open set in (X, τ, A). The concept of soft topology and its applications remain a hot research area [1,2,6–30]. 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 [32–39]. In recent years, the θ_ω-closure operator and θ_ω-open sets in TSs were defined in [3], and their research was continued in [40–43]. The notions of the soft θ-closure operator and soft θ-open sets in STSs were defined in [44]. In this work, the soft θ_ω-closure operator is defined as a new soft operator that lies strictly between the usual soft closure and the soft θ-closure. Several sufficient conditions for equivalence between the soft θ_ω-closure and usual soft closure operators, and between the soft θ_ω-closure and soft sets that lies soft θ-closure operators, soft θ-closure operators, are provided. Via the soft θ_ω-closure operator, the soft θ_ω-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 θ_ω-open sets forms a new soft topology. The soft ω-regularity is characterized through both the soft θ_ω-closure operator and soft θ_ω-open sets. The soft product theorem and several soft mapping theorems are introduced. The 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 θ_ω-open sets of STS and their generated TSs, and vice versa, are studied. In addition to these, the soft θ_ω-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, G ∈ SS(X,A) is defined by

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

(b) [1] G(a) = Y for all a ∈ A is denoted by C_Y.

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

Definition 1.2 [46]

Let G ∈ SS(X,A) and a_x ∈ SP(X, A). Then, a_x is said to belong to F (notation: a_x ∊̃ G) if a_x ⊆̃ G, or equivalently, a_x ∊̃ G if and only if x ∈ G(a).

Theorem 1.3 [5]

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

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 {ℑ_a: a ∈ A} 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 F ∈ SS (X,A) and G ∈ SS (Y,B). Then, the soft Cartesian product of F and G is a soft set denoted by F × G ∈ SS (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 ℬ = {F ×G: F ∈ τ and G ∈ σ}. Subsequently, the soft topology on X × Y relative to A × B that has ℬ as a soft base is known as the product soft topology and is denoted by τ * σ.

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

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 a_x ∈ SP(X,A), there exists G ∈ τ such that a_x ∊̃ G, and G is 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 a_x ∊̃ 1_A − M, there exist F ∈ τ and G ∈ τ_ω such that a_x ∊̃ F, M ⊆̃ G, and F ∩̃ G = 0_A.

2. Soft θ_ω-Closure Operator

Go to

Abstract
1. Introduction
2. Soft θ_ω-Closure Operator
3. Soft θ_ω-Continuity
4. Conclusion
Conflict of Interest
References
Biography

In this section, we introduce the θ_ω-closure operator that lies strictly between the usual soft closure and soft θ-closure. We provide sufficient conditions for equivalence between the soft θ_ω-closure and usual soft closure operators, and between the soft θ_ω-closure and soft θ-closure operators. We use the soft θ_ω-closure operator to define soft θ_ω-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 θ_ω-open sets forms a soft topology. Moreover, we use the soft θ_ω-closure operator and soft θ_ω-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 θ_ω-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 F ∈ SS(X,A).

(a) A soft point a_x ∈ SP(X,A) is in the soft θ-closure of F (a_x ∊̃ Cl_θ(F)) if Cl_τ (G) ∩̃ F ≠ 0_A for any G ∈ τ with a_x ∊̃ G.

(b) F is soft θ-closed if Cl_θ(F) = F.

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

Theorem 2.2 [44]

Let (X, τ,A) be an STS. Then:

(a) (X, τ_θ,A) is an STS.

(b) τ_θ ⊆ τ and τ_θ ≠ τ in general.

The following is the main definition of this work.

Definition 2.3

Let (X, τ,A) be an STS and let F ∈ SS(X, A).

(a) A soft point a_x ∈ SP(X,A) is in the soft θ_ω-closure of F (a_x ∊̃ Cl_θ_{_ω} (F),) if Cl_τ_{_ω} (G) ∩̃ F ≠ 0_A for any G ∈ τ with a_x ∊̃ G.

(b) F is known as soft θ_ω-closed if Cl_θ_{_ω} (F) = F.

(d) The family of all soft θ_ω-open sets in (X, τ,A) is denoted by τ_θ_{_ω}.

Theorem 2.4

Let (X, τ,A) be an STS and let F ∈ SS(X, A). Then:

(a) Cl_τ (F) ⊆̃ Cl_θ_{_ω} (F) ⊆̃ Cl_θ(F).

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

Proof

(a) To demonstrate that Cl_τ (F) ⊆̃ Cl_θ_{_ω} (F), let a_x ∊̃ Cl_τ (F) and let G ∈ τ such that a_x ∊̃ G. As a_x ∊̃ Cl_τ (F), G ∩̃ F ≠ 0_A. Because G ⊆̃ Cl_τ_{_ω} (G), Cl_τ_{_ω} (G) ∩̃ F ≠ 0_A. Thus, a_x ∊̃ Cl_θ_{_ω} (F). To demonstrate that Cl_θ_{_ω} (F) ⊆̃ Cl_θ(F), let a_x ∊̃ Cl_θ_{_ω} (F) and let G ∈ τ such that a_x ∊̃ G. As a_x ∊̃ Cl_θ_{_ω} (F), Cl_τ_{_ω} (G) ∩̃ F ≠ 0_A. Because Cl_τ_{_ω} (G) ⊆̃ Cl_τ (G), Cl_τ (G) ∩̃ F ≠ 0_A. Thus, a_x ∊̃ Cl_θ(F).

(b) Suppose that F is soft θ-closed. Then, Cl_θ (F) = F, and according to (a), Cl_θ_{_ω} (F) = F. Hence, F is soft θ_ω-closed.

(c) Suppose that F is soft θ_ω-closed. Then, Cl_θ_{_ω} (F) = F, and according to (a), Cl_τ (F) = F. Hence, F is soft closed.

Theorem 2.5

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

(a) Cl_τ (F) = Cl_θ_{_ω} (F).

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

Proof

(a) From Theorem 2.4 (a), Cl_τ (F) ⊆̃ Cl_θ_{_ω} (F). To show that Cl_θ_{_ω} (F) ⊆̃ Cl_τ (F), let a_x ∊̃ Cl_θ_{_ω} (F) and G ∈ τ, with a_x ∊̃ G. Then, Cl_τ_{_ω} (G) ∩̃ F ≠ 0_A. Because (X, τ,A) is soft ω-locally indiscrete, Cl_τ_{_ω} (G) = G; hence, G ∩̃ F ≠ 0_A. Therefore, a_x ∊̃ Cl_τ (F).

(b) Suppose that F is soft closed. Then, F = Cl_τ (F), and according to (a), F = Cl_θ_{_ω} (F). Hence, F is soft θ_ω-closed.

Corollary 2.6

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

(a) Cl_τ (F) = Cl_θ_{_ω} (F).

(b) If F is soft closed, F is soft θ_ω-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 F ∈ SS(X,A). Then:

(a) Cl_τ (F) = Cl_θ_{_ω} (F).

(b) If F is soft closed, F is soft θ_ω-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 F ∈ SS(X,A). Then:

(a) Cl_θ(F) = Cl_θ_{_ω} (F).

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

Proof

(a) Using Theorem 2.4(a), Cl_θ_{_ω} (F) ⊆̃ Cl_θ(F). To observe that Cl_θ(F) ⊆̃ Cl_θ_{_ω} (F), let a_x ∊̃ Cl_θ(F) and G ∈ τ, with a_x ∊̃ G. Then, Cl_τ (G) ∩̃ F ≠ 0_A. Because (X, τ,A) is soft anti-locally countable, according to Theorem 14 of [2], Cl_τ_{_ω} (G) = Cl_τ (G); hence, Cl_τ_{_ω} (G) ∩̃ F ≠ 0_A. Therefore, a_x ∊̃ Cl_θ_{_ω} (F).

(b) Suppose that F is soft θ_ω-closed. Then, F = Cl_θ_{_ω} (F), and according to (a), F = Cl_θ(F). Hence, F is soft θ-closed.

Theorem 2.9

For any STS (X, τ,A), τ_θ ⊆ τ_θ_{_ω} ⊆ τ.

Proof

Let F ∈ τ_θ; then, 1_A − F is soft θ-closed. Therefore, using Theorem 2.4(b), 1_A − F is soft θ_ω-closed, and hence, F ∈ τ_θ_{_ω}. Therefore, τ_θ ⊆. τ_θ_{_ω}. To demonstrate that τ_θ_{_ω} ⊆ τ, let F ∈ τ_θ_{_ω}; then, 1_A − F is soft θ_ω-closed. Thus, according to Theorem 2.4(c), 1_A − F is soft closed. Hence, F ∈ τ.

Lemma 2.10

Let (X, τ,A) be an STS. Then, for each F ∈ τ, Cl_θ(F) = Cl_τ (F).

Proof

Let F ∈ τ. Then, using Theorem 2.4(a), Cl_τ (F) ⊆̃ Cl_θ(F). To observe that Cl_θ(F) ⊆̃ Cl_τ (F), let a_x ∊̃ Cl_θ(F) and let G ∈ τ, with a_x ∊̃ G. Then, Cl_τ (G) ∩̃ F ≠ 0_A. Select b_y ∊̃ Cl_τ (G) ∩̃ F. As F ∈ τ, G ∩̃ F ≠ 0_A. It follows that Cl_τ (F) ⊆̃ Cl_θ(F).

Theorem 2.11

Let (X, τ, A) be an STS.

(a) If M ⊆̃ N ⊆̃ 1_A, Cl_θ_{_ω} (M) ⊆̃ Cl_θ_{_ω} (N).

(b) For each M,N ∈ SS(X,A), Cl_θ_{_ω} (M ∪̃ N)= Cl_θ_{_ω} (M) ∪̃ Cl_θ_{_ω} (N).

(d) For each F ∈ τ_ω, Cl_θ_{_ω} (F) = Cl_τ (F).

(e) For each F ∈ τ, Cl_θ(F) = Cl_θ_{_ω} (F) = Cl_τ (F).

Proof

(a) Let a_x ∊̃ Cl_θ_{_ω} (M) and G ∈ τ with a_x ∊̃ G. As a_x ∊̃ Cl_θ_{_ω} (M), Cl_τ_{_ω} (G) ∩̃ M ≠ 0_A. Because M ⊆̃ N, we obtain Cl_τ_{_ω} (G) ∩̃ N ≠ 0_A. This demonstrates that a_x ∊̃ Cl_θ_{_ω} (N).

(b) As M ⊆̃ M ∪̃ N and N ⊆̃ M ∪̃ N, according to (a), Cl_θ_{_ω} (M) ∪̃ Cl_θ_{_ω} (N) ⊆̃ Cl_θ_{_ω} (M ∪̃ N). Conversely, let a_x ∊̃ 1_A−(Cl_θ_{_ω} (M) ∪̃ Cl_θ_{_ω} (N)). Then, there exist G,H ∈ τ such that a_x ∊̃ G ∩̃ H, Cl_τ_{_ω} (G) ∩̃ M = 0_A, and Cl_τ_{_ω} (G) ∩̃ N = 0_A. Therefore, a_x ∊̃ G ∩̃ H ∈ τ and

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

Hence, a_x ∊̃ 1_A − (Cl_θ_{_ω} (M ∪̃ N)).

(c) Let M ∈ SS(X,A). We show that 1_A − Cl_θ_{_ω} (M) ∈ τ. Let a_x ∊̃ 1_A − Cl_θ_{_ω} (M). Then, G ∈ τ exists such that a_x ∊̃ G with Cl_τ_{_ω} (G) ∩̃ M = 0_A; thus, G ∩̃ Cl_θ_{_ω} (M). = 0_A. Hence, 1_A − Cl_θ_{_ω} (M) ∈ τ.

(d) Let F ∈ τ_ω. Then, using Theorem 2.4(a), Cl_τ (F) ⊆̃ Cl_θ_{_ω} (F). To observe that Cl_θ_{_ω} (F) ⊆̃ Cl_τ (F), we assume to the contrary that there exists a_x ∊̃ Cl_θ_{_ω} (F) ∩̃ (1_A − Cl_τ (F)). Because a_x ∊̃ Cl_θ_{_ω}(F) and a_x ∊̃ (1_A − Cl_τ (F)) ∈ τ. Then, Cl_τ_{_ω} (1_A − Cl_τ (F)) ∩̃ F ≠ 0_A. 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) 0_A and 1_A are soft θ_ω-closed sets.

(b) The finite soft union of soft θ_ω-closed sets is soft θ_ω-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 θ_ω-closed sets is soft θ_ω-closed. Let M and N be any two soft θ_ω-closed sets. Then, Cl_θ_{_ω} (M) = M and Cl_θ_{_ω} (N) = N. Thus, according to Theorem 2.11(b),

Clθω (M∪˜N)=Clθ,ω(M)∪˜Clθω N)=M∪˜N.

Hence, M ∪̃ N is soft θ_ω-closed.

(c) Let F_λ be soft θ_ω-closed for all λ ∈ Γ. Then, for all λ ∈ Γ, F_λ = Cl_θ_{_ω} (F_λ). We show that Cl_θ_{_ω} ( ∩̃ {F_λ: λ ∈ Γ}) ⊆̃∩̃ {F_λ: λ ∈ Γ}. Let a_x ∊̃ Cl_θ_{_ω} ( ∩̃ {F_λ: λ ∈ Γ}) and let G ∈ τ, with a_x ∊̃ G; then, Cl_τ_{_ω} (G) ∩̃ ( ∩̃ {F_λ: λ ∈ Γ}) ≠ 0_A. Thus, for every λ ∈ Γ, Cl_τ_{_ω} (G) ∩̃ F_λ ≠ 0_A. Hence, a_x ∊̃ Cl_θ_{_ω} (F_λ) = F_λ for every λ ∈ Γ. It follows that a_x ∊̃∩̃ {F_λ: λ ∈ Γ}.

Theorem 2.13

For any STS (X, τ,A), (X, τ_θ_{_ω},A) is an STS.

Proof

(1) According to Theorem 2.12(a), 0_A and 1_A are soft θ_ω-closed sets, and thus, 0_A, 1_A ∈ τ_θ_{_ω}.

(2) Let M,N ∈ τ_θ_{_ω}; then, 1_A − M and 1_A − N are soft θ_ω-closed sets. Note that

1A-(M∩˜N)=(1A-M)∪˜(1A-N).

Then, using Theorem 2.12(b), 1_A − (M ∩̃ N) is soft θ_ω-closed. Thus, M ∩̃ N ∈ τ_θ_{_ω}.

(3) Let F_λ ∈ τ_θ_{_ω} for every λ ∈ Γ. Then, {1_A − F_λ: λ ∈ Γ} is a family of soft θ_ω-closed sets. Thus, according to Theorem 2.12(c),

1A-∪˜ {Fλ:λ∈Γ}=∩˜{1A-Fλ:λ∈Γ}

is soft θ_ω-closed. Therefore, ∪̃ {F_λ: λ ∈ Γ} ∈ τ_θ_{_ω}.

Theorem 2.14

Let (X, τ,A) be an STS and F ∈ SS(X,A). Then, F ∈ τ_θ_{_ω} if and only if, for each a_x ∊̃ F, there exists G ∈ τ such that a_x ∊̃ G ⊆̃ Cl_τ_{_ω} (G) ⊆̃ F.

Proof

Necessity

Let F ∈ τ_θ_{_ω}, and let a_x ∊̃ F. Then, we determine that 1_A − F is soft θ_ω-closed and a_x ∊̃ 1_A − (1_A − F). As 1_A − F is soft θ_ω-closed, Cl_θ_{_ω} (1_A − F) = 1_A − F, and thus, a_x ∊̃ 1_A − Cl_θ_{_ω} (1_A − F). Hence, there exists G ∈ τ such that a_x ∊̃ G and Cl_τ_{_ω} (G) ∩̃ (1_A − F) = 0_A. Thus, a_x ∊̃ G ⊆̃ Cl_τ_{_ω} (G) ⊆̃ F.

Sufficiency

Suppose that for each a_x ∊̃ F, there exists G ∈ τ such that a_x ∊̃ G ⊆̃ Cl_τ_{_ω} (G) ⊆̃ F. Suppose to the contrary that F ∉ τ_θ_{_ω}. Then, Cl_θ_{_ω}(1_A − F) ≠ 1_A − F, and thus, there exists a_x ∊̃ Cl_θ_{_ω} (1_A − F) − (1_A − F). By assumption, there exists G ∈ τ such that a_x ∊̃ G ⊆̃ Cl_τ_{_ω} (G) ⊆̃ F, and thus, a_x ∊̃ G ∈ τ with Cl_τ_{_ω} (G) ∩̃ (1_A − F) = 0_A. This implies that a_x ∊̃ 1_A − Cl_θ_{_ω} (1_A − F), which is a contradiction.

Corollary 2.15

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

Proof

Let (X, τ,A) be an STS, and let F be a soft open and soft ω-closed set. Let a_x ∊̃ F. Because F is soft ω-closed, Cl_τ_{_ω} (F) = F. Select G = F; then, G ∈ τ and a_x ∊̃ G = Cl_τ_{_ω} (G) = F ⊆̃ F. Thus, according to Theorem 2.14, F is soft θ_ω-open.

Corollary 2.16

Every countable soft open set in an STS is soft θ_ω-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 = τ_θ_{_ω}.

Proof

(a) ⇒ (b): Suppose that (X, τ,A) is soft ω-regular. To observe that τ ⊆ τ_θ_{_ω}, let F ∈ τ and let a_x ∊̃ F. Because (X, τ,A) is soft ω-regular, according to Theorem 13 of [6], there exists G ∈ τ such that a_x ∊̃ G ⊆̃ Cl_τ_{_ω}(G) ⊆̃ F. Therefore, using Theorem 2.14, we obtain F ∈ τ_θ_{_ω}. However, from Theorem 2.9, we obtain τ_θ_{_ω} ⊆ τ.

(b) ⇒ (c): Suppose that τ = τ_θ_{_ω} and let F ∈ SS(X,A). To verify that Cl_θ_{_ω} (F), ⊆̃ Cl_τ (F), let a_x ∊̃ 1_A − Cl_τ (F). Because 1_A − Cl_τ (F) ∈ τ, using (b), 1_A − Cl_τ (F) ∈ τ_θ_{_ω}. Thus, according to Theorem 2.14, there exists G ∈ τ such that a_x ∊̃ G ⊆̃ Cl_τ_{_ω} (G) ⊆̃ 1_A − Cl_τ (F). Therefore, we obtain a_x ∊̃ G ∈ τ and Cl_τ_{_ω} (G) ∩̃ F ⊆̃ Cl_τ_{_ω} (G) ∩̃ Cl_τ (F) = 0_A. Hence, a_x ∊̃ 1_A − Cl_θ_{_ω} (F). However, using Theorem 2.4(a), we obtain Cl_τ (F) ⊆̃ Cl_θ_{_ω} (F).

(c) ⇒ (a): Suppose that Cl_θ_{_ω} (F) = Cl_τ (F) for every F ∈ SS(X,A). Let F ∈ τ and let a_x ∊̃ F. Then, 1_A − F is soft closed and using (c), Cl_θ_{_ω} (1_A − F) = Cl_τ (1_A − F) = 1_A − F. Therefore, a_x ∊̃ 1_A − Cl_θ_{_ω} (1_A − F), and thus, there exists G ∈ τ such that a_x ∊̃ G and Cl_τ_{_ω} (G) ∩̃ (1_A − F) = 0_A. Hence, a_x ∊̃ G ⊆̃ Cl_τ_{_ω} (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 e ∈ A}.

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 F ∈ SS(X, A). Then, for all a ∈ A, Cl_τ_{_a} (F (a)) ⊆ (Cl_τ (F)) (a).

Theorem 2.20

Let (X, τ,A) be an STS. If F ∈ τ_θ, F(a) ∈ (τ_a)_θ for all a ∈ A.

Proof

Suppose that F ∈ τ_θ and let a ∈ A. Let x ∈ F(a) exists G ∈ τ such that a_x ∊̃ G ⊆̃ Cl_τ (G) ⊆̃ F. Thus, we obtain G(a) ∈ τ_a, and using Lemma 2.19, x ∈ G(a) ⊆ Cl_τ_{_a} (G(a)) ⊆ (Cl_τ (G)) (a) ⊆ F(a). Therefore, F(a) ∈ (τ_a)_θ.

Theorem 2.21

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

Proof

Necessity

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

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

Sufficiency

Suppose that F(a) ∈ (ℑ_a)_θ for all a ∈ A. Let a_x ∊̃ F; then, x ∈ F(a) ∈ (ℑ_a)_θ. Therefore, there exists V ∈ ℑ_a such that x ∈ V ⊆ Cl_ℑ_{_a} (V ) ⊆ F(a). Thus, we obtain a_x ∊̃ a_V ⊆̃ a_Cl_{_ℑ_{_a} (}_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) ∈ ℑ_θ for all a ∈ A.

Proof

For each a ∈ A, set ℑ_a = 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 F ∈ τ_θ_{_ω}, F(a) ∈ (τ_a)_θ_{_ω} for all a ∈ A.

Proof

Suppose that F ∈ τ_θ_{_ω} and let a ∈ A. Let x ∈ F(a); then, a_x ∊̃ F. As F ∈ τ_θ_{_ω}, there exists G ∈ τ such that a_x ∊̃ G ⊆̃ Cl_τ_{_ω} (G) ⊆̃ F. Thus, we obtain G(a) ∈ τ_a, and according to Lemma 2.19 and Theorem 7 of [2], x ∈ G(a) ⊆ Cl₍_τ_{_a}₎_{_ω,} (G(a)) = Cl₍_τ_{_ω}₎_{_a} (G(a)) ⊆ (Cl_τ_{_ω} (G)) (a) ⊆ F(a). Therefore, F(a) ∈ (τ_a)_θ_{_ω}.

Theorem 2.24

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

Proof

Necessity

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

Sufficiency

Suppose that F(a) ∈ (ℑ_a)_θ_{_ω} for all a ∈ A. Let a_x ∊̃ F; then, x ∈ F(a) ∈ (ℑ_a)_θ_{_ω}. Thus, there exists V ∈ ℑ_a such that x ∈ V ⊆ Cl_(ℑ_{_a}₎_{_ω} (V ) ⊆ F(a). Thus, we have a_x ∊̃ a_V ⊆̃ a_Cl_{₍_ℑ_{_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))_θ_{_ω} if and only if F(a) ∈ ℑ_θ_{_ω} for all a ∈ A.

Proof

For each a ∈ A, set ℑ_a = 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 F ∈ SS(X,A) and G ∈ SS(Y,B). If F ×G ∈ (τ * σ)_θ_{_ω}, F ∈ τ_θ_{_ω} and G ∈ σ_θ_{_ω}.

Proof

Let a_x ∊̃ F and b_y ∊̃ G, then (a, b)₍_x,y₎ ∊̃ F × G. Because F × G ∈ (τ * σ)_θ_{_ω}, there exists M ∈ τ * σ such that (a, b)₍_x,y₎ ∊̃ M ⊆̃ Cl₍_τ_*_σ₎_{_ω} (M) ⊆̃ F × G. Select K ∈ τ and N ∈ σ such that (a, b)₍_x,y₎ ∊̃ K × N ⊆̃ M. Then, according to Proposition 3(b) of [6], (a, b)₍_x,y₎ ∊̃ K × N ⊆̃ Cl_τ_{_ω} (K) × Cl_σ_{_ω} (N) ⊆̃ Cl₍_τ_*_σ₎_{_ω} (K × N) ⊆̃ Cl₍_τ_*_σ₎_{_ω} (M) ⊆̃ F × G.

Thus, we obtain a_x ∊̃ K ⊆̃ Cl_τ_{_ω} (K) ⊆̃ F and b_y ∊̃ N ⊆̃ Cl_τ_{_ω} (N) ⊆̃ G. Therefore, F ∈ τ_θ_{_ω} and G ∈ σ_θ_{_ω}.

Question 2.27

Let (X, τ,A) and (Y, σ,B) be two STSs. Let F ∈ τ_θ_{_ω} and G ∈ σ_θ_{_ω}. Is it true that F × G ∈ (τ * σ)_θ_{_ω} ?

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

Example 2.28

Let τ = {F ∈ SS(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}), (f_pu(F)) (a) ⊆ ℤ. Thus, f_pu: (X, τ, {a}) → (X, τ_ω, {a}) is soft closed. Because 1_A is soft closed in (X, τ, {a}) but f_pu (1_A) is not soft closed in (X, τ, {a}), f_pu: (X, τ, {a}) → (X, τ, {a}) is not soft closed.

Theorem 2.29

Let f_pu: (X, τ,A) → (Y, σ,B) be a soft function. If f_pu: (X, τ,A) → (Y, σ,B) is soft open and f_pu: (X, τ,A) → (Y, σ_ω,B) is soft closed, f_pu: (X, τ_θ,A) → (Y, σ_θ_{_ω},B) is soft open.

Proof

Let G ∈ τ_θ and let b_y ∊̃ f_pu (G). Select a_x ∊̃ G such that b_y = f_pu (a_x). As a_x ∊̃ G ∈ τ_θ, there exists M ∈ τ such that a_x ∊̃ M ⊆̃ Cl_τ (M) ⊆̃ G. Therefore, b_y = f_pu (a_x) ∊̃ f_pu(M) ⊆̃ f_pu (Cl_τ (M)) ⊆̃ f_pu (G). Because f_pu: (X, τ,A) → (Y, σ,B) is soft open, f_pu(M) ∈ σ. As f_pu: (X, τ,A) → (Y, σ_ω,B) is soft closed, f_pu (Cl_τ (M)) is soft ω-closed in (Y, σ,B), and thus, Cl_τ_{_ω} (f_pu(M)) ⊆̃ f_pu (Cl_τ (M)). Therefore, f_pu (G) ∈ σ_θ_{_ω}. It follows that f_pu: (X, τ_θ,A) → (Y, σ_θ_{_ω},B) is soft open.

Theorem 2.30

Let f_pu: (X, τ,A) → (Y, σ,B) be a soft function. If f_pu: (X, τ,A) → (Y, σ,B) is soft open and f_pu: (X, τ_ω,A) → (Y, σ_ω,B) is soft closed, f_pu: (X, τ_θ_{_ω},A) → (Y, σ_θ_{_ω},B) is soft open.

Proof

Let G ∈ τ_θ_{_ω} and let b_y ∊̃ f_pu (G). Select a_x ∊̃ G such that b_y = f_pu (a_x). As a_x ∊̃ G ∈ τ_θ_{_ω}, there exists M ∈ τ such that a_x ∊̃ M ⊆̃ Cl_τ_{_ω} (M) ⊆̃ G. Thus, b_y = f_pu (a_x) ∊̃ f_pu(M) ⊆̃ f_pu (Cl_τ_{_ω} (M)) ⊆̃ f_pu (G). Because f_pu: (X, τ,A) → (Y, σ, B) is soft open, f(M) ∈ σ. As f_pu: (X, τ_ω,A) → (Y, σ_ω,B) is soft closed, f_pu (Cl_τ_{_ω} (M)) is ω-closed in (Y, σ,B); therefore, Cl_τ_{_ω} (f_pu(M)) ⊆̃ f_pu(Cl_τ_{_ω} (M)) ⊆̃ f_pu (G). It follows that f_pu (G) ∈ σ_θ_{_ω}.

Theorem 2.31

Let f_pu: (X, τ,A) → (Y, σ,B) be a soft function. If both f_pu: (X, τ,A) → (Y, σ,B) and f_pu: (X, τ_ω,A) → (Y, σ_ω,B) are soft continuous, f_pu: (X, τ_θ_{_ω},A) → (Y, σ_θ_{_ω},B) is soft continuous.

Proof

Let H ∈ σ_θ_{_ω} and let ax∈˜fpu-1(H). Then, f_pu (a_x) ∊̃ H; thus, we determine M ∈ σ such that f_pu (a_x) ∊̃ M ⊆̃ Cl_τ_{_ω} (M) ⊆̃ H. Therefore, ax∈˜fpu-1(M)⊆˜fpu-1(Clτω(M))⊆˜fpu-1(H). As f_pu: (X, τ,A) → (Y, σ,B) is soft continuous, fpu-1(M)∈τ. Because f_pu: (X, τ_ω,A) → (Y, σ_ω,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 f_pu: (X, τ_θ_{_ω},A) → (Y, σ_θ_{_ω},B) is soft continuous.

3. Soft θ_ω-Continuity

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Abstract
1. Introduction
2. Soft θ_ω-Closure Operator
3. Soft θ_ω-Continuity
4. Conclusion
Conflict of Interest
References
Biography

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

Definition 3.1 [44]

A soft function f_pu: (X, τ,A) → (Y, σ,B) is said to be soft θ-continuous if for every a_x ∈ SP(X,A) and every G ∈ σ such that f_pu(a_x) ∊̃ G, there exists H ∈ τ such that a_x ∊̃ H and f_pu(Cl_τ (H)) ⊆̃ Cl_σ(G).

Theorem 3.2

Let p: (X, ℑ) → (Y, ℵ) be a function between two TSs and let u: A → B be a function between two sets of parameters. Then, f_pu: (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 f_pu: (X, τ(ℑ),A) → (Y, τ(ℵ),B) is soft θ-continuous. Let x ∈ X and V ∈ ℵ such that p(x) ∈ V. Select a ∈ A; then, f_pu(a_x) = u(a)_p₍_x₎ ∊̃ u(a)_V. As f_pu: (X, τ(ℑ),A) → (Y, τ(ℵ),B) is soft θ-continuous, there exists H ∈ τ such that a_x ∊̃ H and f_pu(Cl_τ₍_ℑ₎(H)) ⊆̃ Cl_τ_(ℵ)(u(a)_V ). Thus, p((Cl_τ₍_ℑ₎(H))(a)⊆(f_pu(Cl_τ₍_ℑ₎(H))) (u(a)) ⊆ (Cl_τ_(ℵ)(u(a)_V ))(u(a)). According to Lemma 2.19, p(Cl_ℑ(H(a))) = p(Cl₍_τ₍_ℑ₎₎_{_a} (H(a))) ⊆ p((Cl_τ₍_ℑ₎(H))(a)). However, using Proposition 2 of [6], (Cl_τ_(ℵ)(u(a)_V )) (u(a)) = u(a)_Cl_{_ℵ}₍_V ₎(u(a)) = Cl_ℵ(V ). Therefore, we obtain x ∈ H(a) ∈ ℑ and p(Cl_ℑ(H(a))) ⊆ Cl_ℵ(V ). Hence, p: (X, ℑ) → (Y, ℵ) is θ-continuous.

Sufficiency

Suppose that p: (X, ℑ) → (Y, ℵ) is θ-continuous. Let a_x ∈ SP(X,A) and let G ∈ τ (ℵ) such that f_pu (a_x) = 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_ℑ(S)) ⊆ Cl_ℵ(G(u(a))). According to Lemma 2.19, Cl_ℵ(G(u(a))) = Cl₍_τ_(ℵ))_{_u₍_a₎} (G(u(a))) ⊆ (Cl_ℵ(G)) (u(a)). Furthermore, from Proposition 2 of [6], Cl_τ₍_ℑ₎ (a_S) = a_Cl_{_ℑ}₍_S₎, and thus, f_pu(Cl_τ₍_ℑ₎(a_S)) = f_pu(a_Cl_{_ℑ}₍_S₎) = u(a)_p₍_Cl_{_ℑ}₍_S₎₎. Therefore, we obtain a_x ∊̃ a_S ∈ τ (ℑ) and f_pu (Cl_τ₍_ℑ₎(a_S)) ⊆̃ Cl_τ_(ℵ)(G). Hence, f_pu: (X, τ( ℑ),A) → (Y, τ(ℵ),B) is soft θ-continuous.

Definition 3.3

A soft function f_pu: (X, τ,A) → (Y, σ,B) is said to be soft θ_ω-continuous if for every a_x ∈ SP(X,A) and every G ∈ σ such that f_pu(a_x) ∊̃ G, there exists H ∈ τ such that a_x ∊̃ H and f_pu(Cl_τ (H)) ⊆̃ Cl_σ_{_ω} (G).

Theorem 3.4

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

Proof

Necessity

Suppose that f_pu: (X, τ(ℑ),A) → (Y, τ(ℵ),B) is soft θ_ω-continuous. Let x ∈ X and V ∈ ℵ such that p(x) ∈ V. Select a ∈ A. Then, f_pu(a_x) = u(a)_p₍_x₎ ∊̃ u(a)_V. As f_pu: (X, τ(ℑ),A) → (Y, τ(ℵ),B) is soft θ_ω-continuous, there exists H ∈ τ such that a_x ∊̃ H and f_pu(Cl_τ₍_ℑ₎(H)) ⊆̃ Cl₍_τ_(ℵ))_{_ω} (u(a)_V ). Thus, p((Cl_τ₍_ℑ₎(H))(a)) ⊆ (f_pu(Cl_τ₍_ℑ₎(H)))(u(a)) ⊆ (Cl_τ_(ℵ)(u(a)_V ))(u(a)). According to Lemma 2.19, p(Cl_ℑ(H(a))) = p(Cl₍_τ₍_ℑ₎₎_{_a} (H(a))) ⊆ p((Cl_τ₍_ℑ₎(H))(a)). In contrast, using Proposition 2 of [6], (Cl₍_τ_(ℵ))_{_ω} (u(a)_V ))(u(a)) = u(a)_Cl_{_ℵ_{_ω}}₍_V ₎(u(a)) = Cl_ℵ_{_ω} (V ). Therefore, we obtain x ∈ H(a) ∈ ℑ and p(Cl_ℑ(H(a))),⊆ Cl_ℵ_{_ω} (V ). Hence, p: (X, ℑ) → (Y, ℵ) is θ_ω-continuous.

Sufficiency

Suppose that p: (X, ℑ) → (Y, ℵ) is θ_ω-continuous. Let a_x ∈ SP(X,A) and let G ∈ τ (ℵ) such that f_pu (a_x) = u (a)_p₍_x₎ ∊̃ G. Then, p(x) ∈ G(u(a)) ∈ ℵ. Because p: (X, ℑ) → (Y, ℵ) is θ_ω-continuous, there exists S ∈ ℑ such that x ∊̃ S and p(Cl_ℑ(S)) ⊆ Cl_ℵ_{_ω} (G(u(a))). According to Lemma 2.19, Cl_ℵ_{_ω} (G(u(a))) = Cl₍_τ_(ℵ_{_ω}₎₎_{_u₍_a₎} (G(u(a))) ⊆ (Cl_ℵ_{_ω} (G)) (u(a)). Furthermore, from Proposition 2 of [6], Cl_τ₍_ℑ₎(a_S) = a_Cl_{_ℑ}₍_S₎, and thus, f_pu(Cl_τ₍_ℑ₎(a_S)) = f_pu(a_Cl_{_ℑ}₍_S₎) = u(a)_p₍_Cl_{_ℑ}₍_S₎₎. Therefore, we obtain a_x ∊̃ a_S ∈ τ (ℑ) and f_pu(Cl_τ₍_ℑ₎(a_S)) ⊆̃ Cl₍_τ_(ℵ))_{_ω} (G). Hence, f_pu: (X, τ (ℑ),A) → (Y, τ (ℵ),B) is soft θ_ω-continuous.

Theorem 3.5

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

Proof

Let f_pu: (X, τ,A) → (Y, σ,B) be soft θ_ω-continuous. Let a_x ∈ SP(X,A). and G ∈ σ such that f_pu(a_x) ∊̃ G. Based on the soft θ_ω-continuity of f_pu, there exists H ∈ τ such that a_x ∊̃ H and f_pu(Cl_τ (H)) ⊆̃ Cl_σ_{_ω} (G). Because Cl_σ_{_ω} (G) ⊆̃ Cl_σ(G), f_pu(Cl_τ (H)) ⊆̃ Cl_σ(G). Therefore, f_pu 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 = {∅︀}, ∪{U ⊆ Y: Y − U is finite }, A = ℝ, and B = {b, c}. Consider the functions p: X → Y and u: A → B, which are defined as p(x) = x for all x ∈ X, 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 θ_ω-continuous. Hence, according to Theorems 3.2 and 3.4, f_pu: (X, τ(ℑ),A) → (Y, τ(ℵ),B) is soft θ-continuous but not soft θ_ω-continuous.

Theorem 3.7

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

Proof

Suppose that f_pu: (X, τ,A) → (Y, σ,B) is a soft θ-continuous function. Let a_x ∈ SP(X,A) and let G ∈ σ such that f_pu(a_x) ∊̃ G. Then, according to the soft θ-continuity of f_pu, there exists H ∈ τ such that a_x ∊̃ H and f_pu(Cl_τ (H)) ⊆̃ Cl_σ(G). As (Y, σ,B) is soft anti-locally countable, according to Theorem 14 of [2], Cl_σ(G) = Cl_σ_{_ω} (G), and thus, f_pu(Cl_τ (H)) ⊆̃ Cl_σ_{_ω} (G). Therefore, f_pu is soft θ_ω-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 θ_ω-continuity are independent concepts:

Example 3.9

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

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

Theorem 3.11

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

Proof

Suppose that f_pu: (X, τ,A) → (Y, σ,B) is a soft θ_ω-continuous function. Let a_x ∈ SP(X,A) and let G ∈ σ such that f_pu(a_x) ∊̃ G. As (Y, σ,B) is soft ω-regular, there exists H ∈ σ such that f_pu(a_x) ∊̃ H ⊆̃ Cl_σ_{_ω} (H) ⊆̃ G. Because f_pu is soft θ_ω-continuous, there exists M ∈ τ such that a_x ∊̃ M and f_pu(Cl_τ (M)) ⊆̃ Cl_σ_{_ω} (H). Therefore, we obtain

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

Hence, f_pu is soft continuous.

Theorem 3.12

Let f_pu: (X, τ,A) → (Y, σ,B) be a soft function such that for every a_x ∈ SP(X,A) and every G ∈ σ such that f_pu(a_x) ∊̃ G, there exists H ∈ τ_θ such that a_x ∊̃ H and f_pu(H) ⊆̃ Cl_σ_{_ω}(G); then, f_pu is soft θ_ω-continuous.

Proof

Let a_x ∈ SP(X,A) and let G ∈ σ such that f_pu(a_x) ∊̃ G. Then, by assumption, there exists H ∈ τ_θ such that a_x ∊̃ H and f_pu(H) ⊆̃ Cl_σ_{_ω} (G). Because a_x ∊̃ H ∈ τ_θ, there exists M ∈ τ such that a_x ∊̃ M ⊆̃ Cl_τ (M) ⊆̃ H. Therefore, we obtain a_x ∊̃ M ∈ τ with f_pu(Cl_τ (M)) ⊆̃ f_pu(H) ⊆̃ Cl_σ_{_ω}(G). Hence, f_pu: (X, τ,A) →.(Y, σ,B) is soft θ_ω-continuous.

Theorem 3.13

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

Proof

Using Theorem 3.12, it is sufficient to prove that for every a_x ∈ SP(X,A) and every G ∈ σ such that f_pu(a_x) ∊̃ G, there exists H ∈ τ_θ such that a_x ∊̃ H and f_pu(H) ⊆̃ Cl_σ_{_ω} (G). Let a_x ∈ SP(X,A) and let G ∈ σ such that f_pu(a_x) ∊̃ G. Set H = f p u - 1 ( C l σ ω ( G ) ). Then, by assumption, H ∈ τ_θ. Furthermore, because f_pu(a_x) ∊̃ 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 f_pu: (X, τ,A) → (Y, σ,B) is soft θ_ω-continuous, for every F ∈ SS(X,A), f_pu(Cl_θ(F)) ⊆̃ Cl_θ_{_ω} (f_pu (F)).

Proof

Suppose that f_pu: (X, τ,A) → (Y, σ,B) is soft θ_ω-continuous and let F ∈ SS(X,A). Let b_y ∊̃ f_pu (Cl_θ(F)). To observe that b_y ∊̃ Cl_θ_{_ω} (f_pu(F)), let G ∈ σ such that b_y ∊̃ G. Because b_y ∊̃ f_pu (Cl_θ(F)), there exists a_x ∊̃ Cl_θ(F) such that b_y = f_pu (a_x). Because f_pu is soft θ_ω-continuous, there exists H ∈ τ such that a_x ∊̃ H and f_pu(Cl_τ (H)) ⊆̃ Cl_σ_{_ω} (G). As a_x ∊̃ H ∈ τ and a_x ∊̃ Cl_θ(F), Cl_τ (H) ∩̃ F ≠ 0_A, and thus, f_pu(Cl_τ (H) ∩̃ F) ≠ f_pu (0_A) = 0_B. Because f_pu(Cl_τ (H) ∩̃ F) ⊆̃ f_pu(Cl_τ(H)) ∩̃ f_pu(F) ⊆̃ Cl_σ_{_ω} (G) ∩̃ f_pu(F), Cl_σ_{_ω}(G) ∩̃ f_pu(F) ≠ 0_B. It follows that b_y ∊̃ Cl_θ_{_ω} (f_pu(F)).

4. Conclusion

Go to

Abstract
1. Introduction
2. Soft θ_ω-Closure Operator
3. Soft θ_ω-Continuity
4. Conclusion
Conflict of Interest
References
Biography

We have defined and investigated the θ_ω-closure operator. Furthermore, we defined soft θ_ω-open sets as a new class of soft sets that form a new soft topology. We have also introduced the soft θ_ω-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 θ_ω-open sets.

Conflict of Interest

Go to

Abstract
1. Introduction
2. Soft θ_ω-Closure Operator
3. Soft θ_ω-Continuity
4. Conclusion
Conflict of Interest
References
Biography

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

References

Go to

Abstract
1. Introduction
2. Soft θ_ω-Closure Operator
3. Soft θ_ω-Continuity
4. Conclusion
Conflict of Interest
References
Biography

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Biography

Go to

Abstract
1. Introduction
2. Soft θ_ω-Closure Operator
3. Soft θ_ω-Continuity
4. Conclusion
Conflict of Interest
References
Biography

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

Abstract

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

1. Introduction

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, G ∈ SS(X,A) is defined by

(a) [1] $G (a) = {\begin{array}{l} Y, & if a = e, \\ \emptyset, & if a \neq e \end{array}$ is denoted by e_Y.

(b) [1] G(a) = Y for all a ∈ A is denoted by C_Y.

(c) [46] $G (a) = {\begin{array}{l} {x}, & if a = e, \\ \emptyset, & if a \neq e \end{array}$ is denoted by e_x 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 G ∈ SS(X,A) and a_x ∈ SP(X, A). Then, a_x is said to belong to F (notation: a_x ∊̃ G) if a_x ⊆̃ G, or equivalently, a_x ∊̃ G if and only if x ∈ G(a).

Theorem 1.3 [5]

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

Theorem 1.4 [47]

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

{F \in S S (X, A) : F (a) \in ℑ for all a \in 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 {ℑ_a: a ∈ A} be an indexed family of topologies on X and let

τ = {F \in S S (X, A) : F (a) \in ℑ_{a} for all a \in A} .

Then, τ defines a soft topology on X relative to A. This soft topology is denoted by $\underset{a \in A}{\oplus} ℑ_{a}$ .

Definition 1.6 [49]

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

(F \times G) ((a, b)) = F (a) \times G (b),

for each (a, b) ∈ A × B.

Definition 1.7 [50]

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

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 a_x ∈ SP(X,A), there exists G ∈ τ such that a_x ∊̃ G, and G is 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 a_x ∊̃ 1_A − M, there exist F ∈ τ and G ∈ τ_ω such that a_x ∊̃ F, M ⊆̃ G, and F ∩̃ G = 0_A.

2. Soft θ_ω-Closure Operator

Definition 2.1 [44]

Let (X, τ,A) be an STS and let F ∈ SS(X,A).

(a) A soft point a_x ∈ SP(X,A) is in the soft θ-closure of F (a_x ∊̃ Cl_θ(F)) if Cl_τ (G) ∩̃ F ≠ 0_A for any G ∈ τ with a_x ∊̃ G.

(b) F is soft θ-closed if Cl_θ(F) = F.

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

Theorem 2.2 [44]

Let (X, τ,A) be an STS. Then:

(a) (X, τ_θ,A) is an STS.

(b) τ_θ ⊆ τ and τ_θ ≠ τ in general.

The following is the main definition of this work.

Definition 2.3

Let (X, τ,A) be an STS and let F ∈ SS(X, A).

(a) A soft point a_x ∈ SP(X,A) is in the soft θ_ω-closure of F (a_x ∊̃ Cl_θ_{_ω} (F),) if Cl_τ_{_ω} (G) ∩̃ F ≠ 0_A for any G ∈ τ with a_x ∊̃ G.

(b) F is known as soft θ_ω-closed if Cl_θ_{_ω} (F) = F.

(d) The family of all soft θ_ω-open sets in (X, τ,A) is denoted by τ_θ_{_ω}.

Theorem 2.4

Let (X, τ,A) be an STS and let F ∈ SS(X, A). Then:

(a) Cl_τ (F) ⊆̃ Cl_θ_{_ω} (F) ⊆̃ Cl_θ(F).

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

Proof

(b) Suppose that F is soft θ-closed. Then, Cl_θ (F) = F, and according to (a), Cl_θ_{_ω} (F) = F. Hence, F is soft θ_ω-closed.

(c) Suppose that F is soft θ_ω-closed. Then, Cl_θ_{_ω} (F) = F, and according to (a), Cl_τ (F) = F. Hence, F is soft closed.

Theorem 2.5

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

(a) Cl_τ (F) = Cl_θ_{_ω} (F).

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

Proof

(b) Suppose that F is soft closed. Then, F = Cl_τ (F), and according to (a), F = Cl_θ_{_ω} (F). Hence, F is soft θ_ω-closed.

Corollary 2.6

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

(a) Cl_τ (F) = Cl_θ_{_ω} (F).

(b) If F is soft closed, F is soft θ_ω-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 F ∈ SS(X,A). Then:

(a) Cl_τ (F) = Cl_θ_{_ω} (F).

(b) If F is soft closed, F is soft θ_ω-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 F ∈ SS(X,A). Then:

(a) Cl_θ(F) = Cl_θ_{_ω} (F).

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

Proof

(b) Suppose that F is soft θ_ω-closed. Then, F = Cl_θ_{_ω} (F), and according to (a), F = Cl_θ(F). Hence, F is soft θ-closed.

Theorem 2.9

For any STS (X, τ,A), τ_θ ⊆ τ_θ_{_ω} ⊆ τ.

Proof

Lemma 2.10

Let (X, τ,A) be an STS. Then, for each F ∈ τ, Cl_θ(F) = Cl_τ (F).

Proof

Theorem 2.11

Let (X, τ, A) be an STS.

(a) If M ⊆̃ N ⊆̃ 1_A, Cl_θ_{_ω} (M) ⊆̃ Cl_θ_{_ω} (N).

(b) For each M,N ∈ SS(X,A), Cl_θ_{_ω} (M ∪̃ N)= Cl_θ_{_ω} (M) ∪̃ Cl_θ_{_ω} (N).

(d) For each F ∈ τ_ω, Cl_θ_{_ω} (F) = Cl_τ (F).

(e) For each F ∈ τ, Cl_θ(F) = Cl_θ_{_ω} (F) = Cl_τ (F).

Proof

\begin{array}{l} C l_{θ_{ω}} (G \tilde{\cap} H) \tilde{\cap} (M \tilde{\cup} N) \\ = (C l_{θ_{ω}} (G \tilde{\cap} H) \tilde{\cap} M) \tilde{\cup} (C l_{θ_{ω}} (G \tilde{\cap} H) \tilde{\cap} N) \\ \tilde{\subseteq} (C l_{θ_{ω}} (G) \tilde{\cap} M) \tilde{\cup} (C l_{θ_{ω}} (H) \tilde{\cap} N) \\ = 0_{A} \tilde{\cup} 0_{A} = 0_{A} . \end{array}

Hence, a_x ∊̃ 1_A − (Cl_θ_{_ω} (M ∪̃ N)).

\begin{array}{l} C l_{τ_{ω}} (1_{A} - C l_{τ} (F)), \tilde{\cap} F \tilde{\subseteq} (1_{A} - C l_{τ} (F)) \tilde{\cap} F \\ \tilde{\subseteq} (1_{A} - F) \tilde{\cap} F = 0_{A} . \end{array}

This is a contradiction.

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

Theorem 2.12

Let (X, τ,A) be an STS. Then:

(a) 0_A and 1_A are soft θ_ω-closed sets.

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

Proof

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

\begin{array}{l} C l_{θ_{ω}} (M \tilde{\cup} N) = C l_{θ_{, ω}} (M) \tilde{\cup} C l_{θ_{ω}} N) \\ = M \tilde{\cup} N . \end{array}

Hence, M ∪̃ N is soft θ_ω-closed.

Theorem 2.13

For any STS (X, τ,A), (X, τ_θ_{_ω},A) is an STS.

Proof

(1) According to Theorem 2.12(a), 0_A and 1_A are soft θ_ω-closed sets, and thus, 0_A, 1_A ∈ τ_θ_{_ω}.

(2) Let M,N ∈ τ_θ_{_ω}; then, 1_A − M and 1_A − N are soft θ_ω-closed sets. Note that

1_{A} - (M \tilde{\cap} N) = (1_{A} - M) \tilde{\cup} (1_{A} - N) .

Then, using Theorem 2.12(b), 1_A − (M ∩̃ N) is soft θ_ω-closed. Thus, M ∩̃ N ∈ τ_θ_{_ω}.

(3) Let F_λ ∈ τ_θ_{_ω} for every λ ∈ Γ. Then, {1_A − F_λ: λ ∈ Γ} is a family of soft θ_ω-closed sets. Thus, according to Theorem 2.12(c),

1_{A} - \tilde{\cup} {F_{λ} : λ \in Γ} = \tilde{\cap} {1_{A} - F_{λ} : λ \in Γ}

is soft θ_ω-closed. Therefore, ∪̃ {F_λ: λ ∈ Γ} ∈ τ_θ_{_ω}.

Theorem 2.14

Let (X, τ,A) be an STS and F ∈ SS(X,A). Then, F ∈ τ_θ_{_ω} if and only if, for each a_x ∊̃ F, there exists G ∈ τ such that a_x ∊̃ G ⊆̃ Cl_τ_{_ω} (G) ⊆̃ F.

Proof

Necessity

Sufficiency

Corollary 2.15

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

Proof

Corollary 2.16

Every countable soft open set in an STS is soft θ_ω-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 = τ_θ_{_ω}.

Proof

Remark

The inclusions in Theorem 2.9 are not equalities in general.

Example 2.18

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

Then,

τ_{θ_{ω}} = {F \in S S (X, A) : F (e) \in {\emptyset, ℝ, ℕ} for all e \in A},

and

τ_{θ} = {F \in S S (X, A) : F (e) \in {\emptyset, ℝ} for all e \in A} .

Lemma 2.19 [5]

Let (X, τ,A) be an STS and F ∈ SS(X, A). Then, for all a ∈ A, Cl_τ_{_a} (F (a)) ⊆ (Cl_τ (F)) (a).

Theorem 2.20

Let (X, τ,A) be an STS. If F ∈ τ_θ, F(a) ∈ (τ_a)_θ for all a ∈ A.

Proof

Theorem 2.21

Let X be an initial universe and let A be a set of parameters. Let {ℑ_a: a ∈ A} be an indexed family of topologies on X. Then, $F \in {(\underset{a \in A}{\oplus} ℑ_{a})}_{θ}$ if and only if F(a) ∈ (ℑ_a)_θ for all a ∈ A.

Proof

Necessity

Suppose that $F \in {(\underset{a \in A}{\oplus} ℑ_{a})}_{θ}$ . Then, according to Theorem 2.20, $F (a) \in {({(\underset{a \in A}{\oplus} ℑ_{a})}_{a})}_{θ}$

for all a ∈ A. However, from Theorem 3.7 of [1], ${(\underset{a \in A}{\oplus} ℑ_{a})}_{a} = ℑ_{a}$ for all a ∈ A. This ends the proof.

Sufficiency

Suppose that F(a) ∈ (ℑ_a)_θ for all a ∈ A. Let a_x ∊̃ F; then, x ∈ F(a) ∈ (ℑ_a)_θ. Therefore, there exists V ∈ ℑ_a such that x ∈ V ⊆ Cl_ℑ_{_a} (V ) ⊆ F(a). Thus, we obtain a_x ∊̃ a_V ⊆̃ a_Cl_{_ℑ_{_a} (}_V₎ ⊆̃ F, with $a_{V} \underset{a \in A}{\oplus} ℑ_{a}$ . Furthermore, according to Proposition 2 of [6], $a_{C l_{ℑ_{a}} (V)} = C l_{\underset{a \in A}{\oplus} ℑ_{a}} (a_{V})$ . Hence, $F \in {(\underset{a \in A}{\oplus} ℑ_{a})}_{θ}$ .

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) ∈ ℑ_θ for all a ∈ A.

Proof

For each a ∈ A, set ℑ_a = ℑ. Then, $τ (ℑ) = \underset{a \in A}{\oplus} ℑ_{a}$ . Thus, using Theorem 2.21, we obtain the result.

Theorem 2.23

Let (X, τ,A) be an STS. If F ∈ τ_θ_{_ω}, F(a) ∈ (τ_a)_θ_{_ω} for all a ∈ A.

Proof

Theorem 2.24

Let X be an initial universe and let A be a set of parameters. Let {ℑ_a: a ∈ A} be an indexed family of topologies on X. Then, $F \in {(\underset{a \in A}{\oplus} ℑ_{a})}_{θ_{ω}}$ if and only if F(a) ∈ (ℑ_a)_θ_{_ω} for all a ∈ A.

Proof

Necessity

Suppose that $F \in {(\underset{a \in A}{\oplus} ℑ_{a})}_{θ_{ω}}$ . Then, using Theorem 2.23, $F (a) \in {({(\underset{a \in A}{\oplus} ℑ_{a})}_{a})}_{θ_{ω}}$ for all a ∈ A. However, according to Theorem 3.7 of [1], ${(\underset{a \in A}{\oplus} ℑ_{a})}_{a} = ℑ_{a}$ for all a ∈ A. This ends the proof.

Sufficiency

Suppose that F(a) ∈ (ℑ_a)_θ_{_ω} for all a ∈ A. Let a_x ∊̃ F; then, x ∈ F(a) ∈ (ℑ_a)_θ_{_ω}. Thus, there exists V ∈ ℑ_a such that x ∈ V ⊆ Cl_(ℑ_{_a}₎_{_ω} (V ) ⊆ F(a). Thus, we have a_x ∊̃ a_V ⊆̃ a_Cl_{₍_ℑ_{_a)_ω}}₍_V₎ ⊆̃ F with $a_{V} \in \underset{a \in A}{\oplus} ℑ_{a}$ . Moreover, according to Proposition 2 in [6], $a_{C l_{{(ℑ_{a})}_{ω}} (V)} = C l_{\underset{a \in A}{\oplus} {(ℑ_{a})}_{ω}} (a_{V})$ . Moreover, using Theorem 8 of [2], $\underset{a \in A}{\oplus} {(ℑ_{a})}_{ω} = {(\underset{a \in A}{\oplus} ℑ_{a})}_{ω}$ . Hence, $F \in {(\underset{a \in A}{\oplus} ℑ_{a})}_{θ_{ω}}$ .

Corollary 2.25

Let (X, ℑ) be a TS and let A be a set of parameters. Then, F ∈ (τ (ℑ))_θ_{_ω} if and only if F(a) ∈ ℑ_θ_{_ω} for all a ∈ A.

Proof

For each a ∈ A, set ℑ_a = ℑ. Then, $τ (ℑ) = \underset{a \in A}{\oplus} ℑ_{a}$ . Thus, using Theorem 2.24, we obtain the result.

Theorem 2.26

Let (X, τ, A) and (Y, σ, B) be two STSs. Let F ∈ SS(X,A) and G ∈ SS(Y,B). If F ×G ∈ (τ * σ)_θ_{_ω}, F ∈ τ_θ_{_ω} and G ∈ σ_θ_{_ω}.

Proof

Thus, we obtain a_x ∊̃ K ⊆̃ Cl_τ_{_ω} (K) ⊆̃ F and b_y ∊̃ N ⊆̃ Cl_τ_{_ω} (N) ⊆̃ G. Therefore, F ∈ τ_θ_{_ω} and G ∈ σ_θ_{_ω}.

Question 2.27

Let (X, τ,A) and (Y, σ,B) be two STSs. Let F ∈ τ_θ_{_ω} and G ∈ σ_θ_{_ω}. Is it true that F × G ∈ (τ * σ)_θ_{_ω} ?

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

Example 2.28

Let τ = {F ∈ SS(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) = {\begin{array}{l} x, & if x \in ℤ, \\ 2, & if x \in ℝ - ℤ . \end{array}

Theorem 2.29

Proof

Theorem 2.30

Proof

Theorem 2.31

Proof

Let H ∈ σ_θ_{_ω} and let $a_{x} \tilde{\in} f_{p u}^{- 1} (H)$ . Then, f_pu (a_x) ∊̃ H; thus, we determine M ∈ σ such that f_pu (a_x) ∊̃ M ⊆̃ Cl_τ_{_ω} (M) ⊆̃ H. Therefore, $a_{x} \tilde{\in} f_{p u}^{- 1} (M) \tilde{\subseteq} f_{p u}^{- 1} (C l_{τ_{ω}} (M)) \tilde{\subseteq} f_{p u}^{- 1} (H)$ . As f_pu: (X, τ,A) → (Y, σ,B) is soft continuous, $f_{p u}^{- 1} (M) \in τ$ . Because f_pu: (X, τ_ω,A) → (Y, σ_ω,B) is soft continuous, and $f_{p u}^{- 1} (C l_{τ_{ω}} (M))$ is soft ω-closed in (X, τ,A); thus, $C l_{τ_{ω}} (f_{p u}^{- 1} (M)) \tilde{\subseteq} f_{p u}^{- 1} (C l_{τ_{ω}} (M)) \tilde{\subseteq} f_{p u}^{- 1} (H)$ . Hence, $f_{p u}^{- 1} (H) \in τ_{θ_{ω}}$ . It follows that f_pu: (X, τ_θ_{_ω},A) → (Y, σ_θ_{_ω},B) is soft continuous.

3. Soft θ_ω-Continuity

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

Definition 3.1 [44]

Theorem 3.2

Proof

Necessity

Sufficiency

Definition 3.3

Theorem 3.4

Proof

Necessity

Sufficiency

Theorem 3.5

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

Proof

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

Example 3.6

Theorem 3.7

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

Proof

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 θ_ω-continuity are independent concepts:

Example 3.9

Example 3.10

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

Theorem 3.11

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

Proof

f_{p u} (M) \tilde{\subseteq} f_{p u} (C l_{τ} (M)) \tilde{\subseteq} C l_{σ_{ω}} (H) \tilde{\subseteq} G

Hence, f_pu is soft continuous.

Theorem 3.12

Proof

Theorem 3.13

Let f_pu: (X, τ,A) → (Y, σ,B) be a soft function such that for every G ∈ σ, $f_{p u}^{- 1} (C l_{σ_{ω}} (G)) \in τ_{θ}$ , f_pu is soft θ_ω-continuous.

Proof

Using Theorem 3.12, it is sufficient to prove that for every a_x ∈ SP(X,A) and every G ∈ σ such that f_pu(a_x) ∊̃ G, there exists H ∈ τ_θ such that a_x ∊̃ H and f_pu(H) ⊆̃ Cl_σ_{_ω} (G). Let a_x ∈ SP(X,A) and let G ∈ σ such that f_pu(a_x) ∊̃ G. Set $H = f_{p u}^{- 1} (C l_{σ_{ω}} (G))$ . Then, by assumption, H ∈ τ_θ. Furthermore, because f_pu(a_x) ∊̃ G, $a_{x} \tilde{\in} f_{p u}^{- 1} (G) \tilde{\subseteq} f_{p u}^{- 1} (C l_{σ_{ω}} (G)) = H$ . Moreover, as $H = f_{p u}^{- 1} (C l_{σ_{ω}} (G)), f_{p u} (H) = f_{p u} (f_{p u}^{- 1} (C l_{σ_{ω}} (G))) \tilde{\subseteq} C l_{σ_{ω}} (G)$ .

Theorem 3.14

If f_pu: (X, τ,A) → (Y, σ,B) is soft θ_ω-continuous, for every F ∈ SS(X,A), f_pu(Cl_θ(F)) ⊆̃ Cl_θ_{_ω} (f_pu (F)).

Proof

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Previous Article LIST Next Article

International Journalof Fuzzy Logic andIntelligent Systems

Original Article

Soft -Open Sets and Soft -Continuity

Abstract

1. Introduction

Definition 1.1

Definition 1.2 [46]

Theorem 1.3 [5]

Theorem 1.4 [47]

Theorem 1.5 [1]

Definition 1.6 [49]

Definition 1.7 [50]

Definition 1.8

2. Soft θω-Closure Operator

Definition 2.1 [44]

Theorem 2.2 [44]

Definition 2.3

Theorem 2.4

Theorem 2.5

Corollary 2.6

Corollary 2.7

Theorem 2.8

Theorem 2.9

Lemma 2.10

Theorem 2.11

Theorem 2.12

Theorem 2.13

Theorem 2.14

Corollary 2.15

Corollary 2.16

Theorem 2.17

Remark

Example 2.18

Lemma 2.19 [5]

Theorem 2.20

Theorem 2.21

Corollary 2.22

Theorem 2.23

Theorem 2.24

Corollary 2.25

Theorem 2.26

Question 2.27

Example 2.28

Theorem 2.29

Theorem 2.30

Theorem 2.31

3. Soft θω-Continuity

Definition 3.1 [44]

Theorem 3.2

Sufficiency

Definition 3.3

Theorem 3.4

Sufficiency

Theorem 3.5

Example 3.6

Theorem 3.7

Theorem 3.8 [51]

Example 3.9

Example 3.10

Theorem 3.11

Theorem 3.12

Theorem 3.13

Theorem 3.14

4. Conclusion

Conflict of Interest

References

Biography

Article

Original Article

Soft -Open Sets and Soft -Continuity

Abstract

1. Introduction

Definition 1.1

Definition 1.2 [46]

Theorem 1.3 [5]

Theorem 1.4 [47]

Theorem 1.5 [1]

Definition 1.6 [49]

Definition 1.7 [50]

Definition 1.8

International Journal
of Fuzzy Logic and
Intelligent Systems

2. Soft θ_ω-Closure Operator

3. Soft θ_ω-Continuity

2. Soft θ_ω-Closure Operator

3. Soft θ_ω-Continuity