Soft -Continuity and Soft -Continuity in Soft Topological Spaces

Samer Al Ghour

doi:10.5391/IJFIS.2022.22.2.183

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International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(2): 183-192

Published online June 25, 2022

https://doi.org/10.5391/IJFIS.2022.22.2.183

Soft -Continuity and Soft -Continuity in Soft Topological Spaces

Samer Al Ghour

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

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

Received: November 28, 2021; Revised: January 16, 2022; Accepted: January 26, 2022

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

Go to

Abstract
1. Introduction and Preliminaries
2. Soft ω-Continuity
3. Soft ω_s-Continuity
4. Conclusion
Conflict of Interest
References
Biography

In this study, soft ω-continuity and soft ω_s-continuity are introduced as two new classes of soft functions, and several characterizations of these concepts are given. It is proven that soft ω-continuity is weaker than soft continuity and that soft ω_s-continuity lies strictly between soft ω-continuity and soft semi-continuity. Sufficient conditions are introduced for the equivalence between soft ω_s-continuity and soft ω-continuity, as well as that between soft ω_s-continuity and soft semi-continuity. Furthermore, composition theorems regarding soft ω-continuity and soft ω_s-continuity are given. Finally, the relationships between the generated soft topological spaces and induced topological spaces are studied.

Keywords: Soft semi-continuous functions, Soft ω-open sets, ω-continuity, ω_s-continuity, Soft generated soft topological space, Soft induced topological spaces

1. Introduction and Preliminaries

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Abstract
1. Introduction and Preliminaries
2. Soft ω-Continuity
3. Soft ω_s-Continuity
4. Conclusion
Conflict of Interest
References
Biography

Herein, we follow the notions and terminology that appear in [1–3]. Throughout this paper, ST denotes a topological space and STS denotes a soft topological space. To deal with uncertain objects, Molodtsov [4] introduced soft sets in 1999. Let X be a universal set and A be a set of parameters. A function K : A → ℘(X) is called a soft set over X relative to A. SS (X, A) denotes the family of all soft sets over X relative to A. A soft set K ∈ SS (X, A) is called a countable soft set if for each a ∈ A, the set K (a) is countable. CSS (X, A) denotes the family of all members of SS (X, A) that are countable. The null soft set and absolute soft set are denoted by 0_A and 1_A, respectively. The concept of STS was defined in [5] as follows: the triplet (X, τ, A), where τ ⊆ SS (X, A), is called an STS if 0_A and 1_A ∈ τ , τ are closed under a finite soft intersection, and τ is closed under an arbitrary soft union. For an STS (X, τ, A) the members of τ are called soft open sets, and their soft complements are called soft closed sets. The family of all soft closed sets is denoted by τ^c. Mathematicians have extended many STs to STSs; some recent studies have also been conducted ([1–3,6–29]). Chen [30] introduced the concept of soft semi-open sets. Following this, many research papers on soft semi-open sets and their modifications appeared. Soft ω-openness was defined in [2] as a generalization of soft openness. In a previous study [3], the author used soft ω-open sets to define ω_s-open sets as a class of soft sets that lies strictly between the soft open sets and soft semi-open sets. In this study, soft ω-continuity and soft ω_s-continuity are introduced as two new classes of soft functions using soft ω-open sets and soft ω_s-open sets. In addition, several characterizations of these concepts are given. It is proven that soft ω-continuity is weaker than soft continuity, and soft ω_s-continuity lies strictly between soft ω-continuity and soft semi-continuity. Sufficient conditions for the equivalence between soft ω_s-continuity and soft ω- continuity, as well as that between soft ω_s-continuity and soft semi-continuity, are introduced. Composition theorems regarding soft ω-continuity and soft ω_s-continuity are also provided. Finally, the relationships regarding generated STSs and induced STs are studied.

Let (X, τ, A) be an STS, (X, ℑ) be a TS, H ∈ SS(X, A), and D ⊆ X. Throughout this paper, Cl_τ (H), int_τ (H), and Cl_ℑ (D) denotes the soft closure of H in (X, τ, A), the soft interior of H in (X, τ, A), and the closure of D in (X,ℑ), respectively.

The following definitions and results are used in the following:

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

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

(c) $G (a) = {\begin{array}{l} {x}, & if a = e, \\ \emptyset, & if a \neq e \end{array}$ , is denoted by e_x, and is called a soft point.

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

Definition 1.2 [31]

LetG ∈ SS(X, A) and e_x ∈ SP(X, A). Then e_x is said to belong to F (notation: e_x ∊̃ G) if e_x ∊̃ G or, equivalently, e_x ∊̃ G if and only if x ∈ G(e).

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 [32]

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

An STS (X, τ, A) is called

(a) [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.

(b) [2] soft anti-locally countable if for every F ∈ τ −{0_A}, F is not a countable soft set.

Definition 1.7

Let (X,ℑ) be a ST, and let D ⊆ X. D is called

(a) [33] semi-open if there exists V ∈ ℑ such that V ⊆ D ⊆ Cl_ℑ (V ). SO(X, ℑ) denotes the family of all semi-open sets in (X,ℑ).

(b) [34] ω_s-open if there exists V ∈ ℑ such that V ⊆ D ⊆ Cl_ℑ_ω (V ). ω_s(X, ℑ) denotes the family of all ω_s-open sets in (X, ℑ).

Similar types of generalized sets in ideal STs have been discussed in [35–39].

Definition 1.8 [40]

A function g : (X, ℑ) → (Y, ℵ) is called ω-continuous at a point x ∈ X if for every V ∈ ℵ with g(x) ∈ V, there exists U ∈ ℑ_ω such that x ∈ U and g(U) ⊆ V . If g is ω-continuous at each point x ∈ X, then g is called ω-continuous.

Definition 1.9 [33]

A function g : (X, ℑ) → (Y, ℵ) is called semi-continuous if for each V ∈ ℵ, g⁻¹(V ) ∈ SO(X, ℑ).

Definition 1.10 [34]

A function g : (X, ℑ) → (Y, ℵ) is called ω_s-continuous if for each V ∈ ℵ, g⁻¹(V ) ∈ ω_s(X, ℑ).

Definition 1.11

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

(a) [30] K is called a soft semi-open set in (X, τ, A) if there exists F ∈ τ such that F ⊂̃ K ⊂̃ Cl_τ (F). SO(X, τ, A) denotes the family of all soft semi-open sets in (X, τ, A).

(b) [30] K is called a soft semi-closed set in (X, τ, A) if 1_A − K ∈ SO(X, τ, A).

(c) [3] K is called a soft ω_s-open set in (X, τ, A) if there exists F ∈ τ such that F ⊂̃ K ⊂̃ Cl_τω (F). ω_s(X, τ, A) denotes the family of all soft ω_s-open sets in (X, τ, A).

According to Definition 1.11, a soft semi-open set always contains a soft open set. Therefore, this is a particular type of generalized soft open set. One can specify another type of generalized set in the STS that does not necessarily contain a soft open set. In the ST, two different types of sets have been discussed in [41].

Definition 1.12 [42]

A soft function f_pu : (X, τ, A) → (Y, σ, B) is called soft semi-continuous if for every G ∈ σ, $f_{p u}^{- 1} (G) \in S O (X, τ, A)$ .

Definition 1.13 [3]

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

(a) The soft ω_s-closure of H in (X, τ, A) is denoted by ω_s-Cl_τ (H) and defined by

ω_{s} - {C l}_{τ} (H) = \tilde{\cap} {M : M is soft ω_{s} -closed in (X, τ, A) and H \tilde{\subseteq} M} .

(b) The soft ω_s-interior of H in (X, τ,A) is denoted by ω_sint_τ (H) and defined by

ω_{s} - {i n t}_{τ} (H) = \tilde{\cup} {K : K is soft ω_{s} -open in (X, τ, A) and K \tilde{\subseteq} H} .

2. Soft ω-Continuity

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Abstract
1. Introduction and Preliminaries
2. Soft ω-Continuity
3. Soft ω_s-Continuity
4. Conclusion
Conflict of Interest
References
Biography

In this section, we introduce the concept of soft ω-continuous function and establish its main properties. With the help of examples, we investigate the interaction between these concepts within the frameworks of general topology and soft topology.

Definition 2.1

A soft function f_pu : (X, τ,A) → (Y, σ,B) is called soft ω-continuous at a soft point a_x ∈ SP(X,A), if G ∈ σ with f_pu(a_x)∊̃G, there exists H ∈ τ_ω such that a_x∊̃H and f_pu(H)⊂̃G. If f_pu is soft ω-continuous at each soft point a_x ∈ SP(X,A), then f_pu is called soft ω-continuous.

Theorem 2.2

For a soft function f_pu : (X, τ,A) → (Y, σ,B), the following conditions are equivalent:

(a) f_pu is soft ω-continuous.

(b) For each G ∈ σ, $f_{p u}^{- 1} (G) \in τ_{ω}$ .

(d) For a soft subbase of (Y, σ,B), $f_{p u}^{- 1} (S) \in τ_{ω}$ for all .

(e) For each N ∈ σ^c, $f_{p u}^{- 1} (N) \in {(τ_{ω})}^{c}$ .

(f) For each K ∈ SS(X,A), f_pu(Cl_τ_{_ω} (K))⊂̃Cl_σ (f_pu(K)).

(g) For each L ∈ SS(Y,B), ${C l}_{τ_{ω}} (f_{p u}^{- 1} (L)) \tilde{\subseteq} f_{p u}^{- 1} ({C l}_{σ} (L))$ .

Proof

(a) =⇒ (b): Let G ∈ σ and let $a_{x} \tilde{\in} f_{p u}^{- 1} (G)$ . Then f_pu (a_x) ∊̃G. By (a), f_pu is soft ω-continuous at a_x. Thus, there exists H ∈ τ_ω such that a_x∊̃H and f_pu(H)⊂̃G. Because f_pu(H)⊂̃G, then we have $a_{x} \tilde{\in} H \tilde{\subseteq} f_{p u}^{- 1} (f_{p u} (H)) \tilde{\subseteq} f_{p u}^{- 1} (G)$ . Therefore, $f_{p u}^{- 1} (G) \in τ_{ω}$ .

(b) =⇒ (c): Obvious.

(d) ⇒ (e): Let N ∈ σ^c. We will show that $1_{A} - f_{p u}^{- 1} (N) \in τ_{ω}$ . Let $a_{x} \tilde{\in} 1_{A} - f_{p u}^{- 1} (N) = f_{p u}^{- 1} (1_{B} - N)$ . Then f_pu (a_x) ∊̃; 1_B − N ∈ σ. Since in (d) is a soft subbase for (Y, σ,B), then there exist . such that

f_{p u} (a_{x}) \tilde{\in} S_{1} \tilde{\cap} S_{2} \tilde{\cap} \dots \tilde{\cap} S_{n} \tilde{\subseteq} 1_{B} - N .

So,

\begin{array}{l} a_{x} \tilde{\in} f_{p u}^{- 1} (S_{1}) \tilde{\cap} f_{p u}^{- 1} (S_{2}) \tilde{\cap} \dots \tilde{\cap} f_{p u}^{- 1} (S_{n}) \\ \tilde{\subseteq} f_{p u}^{- 1} (1_{B} - N) = 1_{A} - f_{p u}^{- 1} (N) . \end{array}

By (d), $f_{p u}^{- 1} (S_{1}) \tilde{\cap} f_{p u}^{- 1} (S_{2}) \tilde{\cap} \dots \tilde{\cap} f_{p u}^{- 1} (S_{n}) \in τ_{ω}$ , and thus, $1_{A} - f_{p u}^{- 1} (N) \in τ_{ω}$ .

(e) ⇒ (f): Let K ∈ SS(X,A). Then Cl_σ (f_pu(K)) ∈ σ^c. Thus, by (e), $f_{p u}^{- 1} ({C l}_{σ} (f_{p u} (K))) \in {(τ_{ω})}^{c}$ . Because $K \tilde{\subseteq} f_{p u}^{- 1} (f_{p u} (K)) \tilde{\subseteq} f_{p u}^{- 1} ({C l}_{σ} (f_{p u} (K)))$ ,

{C l}_{τ_{ω}} (K) \tilde{\subseteq} f_{p u}^{- 1} ({C l}_{σ} (f_{p u} (K))),

and so,

\begin{array}{l} f_{p u} ({C l}_{τ_{ω}} (K)) \tilde{\subseteq} f_{p u} (f_{p u}^{- 1} ({C l}_{σ} (f_{p u} (K)))) \\ \tilde{\subseteq} {C l}_{σ} (f_{p u} (K)) . \end{array}

(f) ⇒ (g): Let L ∈ SS(Y,B). Then $f_{p u}^{- 1} (L) \in S S (X, A)$ . Thus, by (f),

f_{p u} ({C l}_{τ_{ω}} (f_{p u}^{- 1} (L))) \tilde{\subseteq} {C l}_{σ} (f_{p u} (f_{p u}^{- 1} (L))) \tilde{\subseteq} {C l}_{σ} (L) .

Therefore,

\begin{array}{l} {C l}_{τ_{ω}} (f_{p u}^{- 1} (L)) \tilde{\subseteq} f_{p u}^{- 1} (f_{p u} ({C l}_{τ_{ω}} (f_{p u}^{- 1} (L)))) \\ \tilde{\subseteq} f_{p u}^{- 1} ({C l}_{σ} (L)) . \end{array}

(g) ⇒ (a): Let a_x ∈ SP(X,A) and G ∈ σ such that f_pu(a_x) ∈ G. Then 1_B−G ∈ σ^c. By (g), ${C l}_{τ_{ω}} (1_{A} - f_{p u}^{- 1} (G)) = {C l}_{τ_{ω}} (f_{p u}^{- 1} (1_{B} - G)) \tilde{\subseteq} f_{p u}^{- 1} ({C l}_{σ} (1_{B} - G)) = \tilde{\subseteq} f_{p u}^{- 1} (1_{B} - G) = 1_{A} - f_{p u}^{- 1} (G)$ . Thus, $1_{A} - f_{p u}^{- 1} (G) \in {(τ_{ω})}^{c}$ and hence $f_{p u}^{- 1} (G) \in τ_{ω}$ . Put $H = f_{p u}^{- 1} (G)$ . Then a_x∊̃;H ∈ τ_ω and $f_{p u} (H) = f_{p u} (f_{p u}^{- 1} (G)) \tilde{\subseteq} G$ . Therefore, f is soft ω-continuous at a_x.

Theorem 2.3

If f_pu : (X, τ,A) → (Y, σ,B) is soft ω-continuous at a_x ∈ SP(X,A), then p : (X, τ_a) → (Y, σ_u₍_a₎) is ω-continuous at x.

Proof

Suppose that f_pu : (X, τ,A) → (Y, σ,B) is soft ω-continuous at a_x. Let V ∈ σ_u₍_a₎ such that p (x) ∈ V . Choose G ∈ σ such that G(u(a)) = V . Then p (x) ∈ G(u(a)), so f_pu(a_x) = (u(a))_p₍_x₎ ∈ G. Because f_pu is soft ω-continuous at a_x, there exists H ∈ τ_ω such that a_x∊̃H and f_pu(H)⊂̃G. Thus, we have x ∈ H(a) ∈ (τ_ω)_a. Also, by Theorem 7 of [2], (τ_ω)_a = (τ_a)_ω. To see that p(H(a)) ⊆ V , let z ∈ H (a); then, a_z∊̃H. Because f_pu(H)⊂̃G, f_pu(a_z) = (u (a))_p₍_z₎ ∊̃G and so p(z) ∈ G(u(a)) = V . This shows that p : (X, τ_a) → (Y, σ_u₍_a₎) is ω-continuous at x.

Theorem 2.4

If f_pu : (X, τ,A) → (Y, σ,B) is soft ω-continuous, then p : (X, τ_a) → (Y, σ_u₍_a₎) is ω-continuous for all a ∈ A.

Proof

Let a ∈ A. To demonstrate that p : (X, τ_a) → (Y, σ_u₍_a₎) is ω-continuous, let x ∈ X. Then f_pu : (X, τ,A) → (Y, σ,B) is soft ω-continuous at a_x. So, by Theorem 2.3, p : (X, τ_a) → (Y, σ_u₍_a₎) is ω-continuous at x. This shows that p : (X, τ_a) → (Y, σ_u₍_a₎) is ω-continuous.

Theorem 2.5

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 p : (X, ℑ) → (Y, ℵ) is ω-continuous at a point x ∈ X if and only if f_pu : (X, τ (ℑ),A) → (Y, τ (ℵ),B) is soft ω-continuous at a_x for all a ∈ A.

Proof

Necessity

Suppose that p : (X, ℑ) → (Y, ℵ) is ω-continuous at x. Let a ∈ A and let G ∈ τ (ℵ) such that f_pu(a_x) ∊̃G. Then p(x) ∈ G(u (a)) ∈ ℵ. Because p : (X, ℑ) → (Y, ℵ) is ω-continuous at x, there exists U ∈ ℑ_ω such that x ∈ U and p(U) ⊆ G(u (a)). Thus, we have a_x∊̃a_U ∈ τ (ℑ_ω) = (τ (ℑ))_ω and f_pu(a_U) = (u(a))_p₍_U₎ ⊂̃G. Hence, f_pu : (X, τ (ℑ),A) → (Y, τ (ℵ),B) is soft ω-continuous at a_x.

Sufficiency

Suppose that f_pu : (X, τ(ℑ),A) → (Y, τ(ℵ), B) is soft ω_p-continuous at a_x. Then by Theorem 2.3,

p : (X, (τ (ℑ))_a) → (Y, (τ (ℵ))_u₍_a₎) is ω-continuous at x. Because (τ (ℑ))_a = ℑ and (τ (ℵ))_u₍_a₎ = ℵ, p : (X, ℑ) → (Y, ℵ) is ω-continuous at x.

Corollary 2.6

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 p : (X, ℑ) → (Y, ℵ) is ω-continuous if and only if f_pu : (X, τ (ℑ),A) → (Y, τ (ℵ),B) is soft ω-continuous.

Theorem 2.7

Soft continuous soft functions are soft ω-continuous.

Proof

Follows from Theorem 2.2 and Theorem 2(a) of [2].

The following example shows that the converse of Theorem 2.7 is not true in general:

Example 2.8

Let X = Y = ℤ, and A = B = ℝ. Let ℑ = {∅︀,X,ℕ} and ℵ be the cofinite topology on Y . Define p : X → Y and u : A → B by p (x) = x and u(a) = a for all x ∈ X and a ∈ A. Then f_pu : (X, τ (ℑ),A) → (Y, τ (ℵ),B) is soft ω-continuous but not soft continuous.

Theorem 2.9

If f_pu : (X, τ,A) → (Y, σ,B) is soft ω-continuous and if f_qv : (Y, σ,B) → (Z, δ,C) is soft continuous, then f₍_q_◦_p₎₍_v_◦_u₎ : (X, τ,A) → (Z, δ,C) is soft ω-continuous.

Proof

Let K ∈ δ. Because f_qv : (Y, σ, ,B) → (Z, δ,C) is soft continuous, then $f_{q u}^{- 1} (K) \in σ$ . Because f_pu is soft ω-continuous, $f_{q u}^{- 1} (f_{q v}^{- 1} (K)) \in τ_{ω}$ . However, $f_{p u}^{- 1} (f_{q v}^{- 1} (K)) = f_{(q \circ p) (v \circ u)}^{- 1} (K)$ . Therefore, $f_{(q \circ p) (v \circ u)}^{- 1} (K) \in τ_{ω}$ . Hence, f₍_q_◦_p₎₍_v_◦_u₎ : (X, τ,A) → (Z, δ,C) is soft ω-continuous.

Theorem 2.10

If f_pu : (X, τ,A) → (Y, σ,B) is soft ω-continuous and Z is a non-empty subset of X, then the soft restriction f(_p_{_{|_Z}})_u : (Z, τ_Z,A) → (Y, σ,B) is soft ω-continuous.

Proof

Let K ∈ σ. Since f_pu : (X, τ,A) → (Y, σ,B) is soft ω-continuous, then $f_{p u}^{- 1} (K) \in τ_{ω}$ . So, $f_{(p_{∣ Z}) u}^{- 1} (K) = f_{p u}^{- 1} (K) \tilde{\cap} C_{Y} \in {(τ_{ω})}_{Y}$ . Hence, by Theorem 15 of [2], $f_{(p_{∣ Z}) u}^{- 1} (K) \in {(τ_{Y})}_{ω}$ . Therefore, f(_p_{_{|_Z}})_u : (Z, τ_Z,A) → (Y, σ,B) is soft ω-continuous.

Theorem 2.11

Let f_pu : (X, τ,A) → (Y, σ,B) be a soft function and let 1_A = C_Z ∪̃C_W, where C_Z, C_W ∈ τ^c − {0_A}. If f(_p_{_{|_Z}})_u : (Z, τ_Z,A) → (Y, σ,B) and f(_p_{_{|_W}})_u : (W, τ_W,A) → (Y, σ,B) are soft ω-continuous functions, then f_pu is soft ω-continuous.

Proof

We apply statement (e) of Theorem 2.2. Let N ∈ σ^c. Then

\begin{array}{l} f_{p u}^{- 1} (N) = f_{p u}^{- 1} (N) \tilde{\cap} 1_{A} \\ = f_{p u}^{- 1} (N) \tilde{\cap} (C_{Z} \tilde{\cup} C_{W}) \\ = (f_{p u}^{- 1} (N) \tilde{\cap} C_{Z}) \tilde{\cup} (f_{p u}^{- 1} (N) \tilde{\cap} C_{W}) \\ = (f_{(p_{∣ Z}) u}^{- 1} (N)) \tilde{\cup} (f_{(p_{∣ W}) u}^{- 1} (N)) . \end{array}

Because f(_p_{_{|_Z}})_u and f(_p_{_{|_W}})_u are soft ω-continuous, then

f_{(p_{∣ Z .}) u}^{- 1} (N) \in {({(τ_{Z})}_{ω})}^{c} and f_{(p_{∣ W .}) u}^{- 1} (N) \in {({(τ_{W})}_{ω})}^{c} .

Therefore, by Theorem 17 of [2], $f_{(p_{∣ Z .}) u}^{- 1} (N) \in {({(τ_{ω})}_{Z})}^{c}$ and $f_{(p_{∣ W .}) u}^{- 1} (N) \in {({(τ_{ω})}_{W})}^{c}$ . Since C_Z, C_W ∈ τ ^c, then $f_{(p_{∣ Z .}) u}^{- 1} (N), f_{(p_{∣ W .}) u}^{- 1} (N) \in {(τ_{ω})}^{c}$ . Therefore, $f_{p u}^{- 1} (N) \in {(τ_{ω})}^{c}$ .

Theorem 2.12

Let f_pu : (X, τ,A) → (Y, σ,B) be a soft function and let a_x ∈ SP(X,A). If there exists Z ⊆ X such that C_Z ∈ τ_ω such that a_x∊̃C_Z and f(_p_{_{|_Z}})_u : (Z, τ_Z,A) → (Y, σ,B) is soft ω-continuous at a_x, then f_pu : (X, τ,A) → (Y, σ,B) is soft ω-continuous at a_x.

Proof

Let G ∈ σ such that f_pu(a_x) ∈ G. Because f(_p_{_{|_Z}})_u is soft ω-continuous at a_x, there exists H ∈ (τ_Z)_ω such that a_x∊̃H and f(_p_{_{|_Z}})_u(H) ⊂̃G. BecauseH ∈ (τ_Z)_ω and C_Z ∈ τ_ω, H ∈ τ_ω. Therefore, f_pu : (X, τ,A) → (Y, σ,B) is soft ω-continuous at a_x.

Corollary 2.13

f_pu : (X, τ, A) → (Y, σ, B) be a soft function. Let {C_X_{_α} : α ∈ Δ} ⊆ τ such that 1_A = ∪̃{C_X_{_α} : α ∈ Δ}. If for each α ∈ Δ, f(_p_|_X_{_α})^u : (X_α, τ_X_{_α}, A) → (Y, σ, B) is soft ω-continuous, then f_pu : (X, τ, A) → (Y, σ, B) is soft ω-continuous.

Proof

Let a_x ∈ SP(X, A). We show that f_pu : (X, τ, A) → (Y, σ, B) is soft ω-continuous at a_x. Because 1_A = ∪̃{C_X_{_α} : α ∈ Δ}, then there exists α_∘ ∈ Δsuch that a_x∊̃C_X_{_α_{_∘}}. Therefore, by Theorem 2.12, it follows that f_pu : (X, τ, A) → (Y, σ, B) is soft ω-continuous.

3. Soft ω_s-Continuity

Go to

Abstract
1. Introduction and Preliminaries
2. Soft ω-Continuity
3. Soft ω_s-Continuity
4. Conclusion
Conflict of Interest
References
Biography

In this section, we introduce the concept of a soft ω_s-continuous function and establish its main properties. We also provide several characterizations of this concept, and study the relationships between soft ω_s-continuity and some other types of soft continuity.

Definition 3.1

A soft function f_pu : (X, τ, A) → (Y, σ, B) is called soft ω_s-continuous if for every G ∈ σ, $f_{p u}^{- 1} (G) \in ω_{s} (X, τ, A)$ .

Theorem 3.2

Soft continuous functions are soft ω_s-continuous.

Proof

Let f_pu : (X, τ, A) → (Y, σ, B) be soft continuous and let G ∈ σ. By soft continuity of f_pu, $f_{p u}^{- 1} (G) \in τ$ . So by Theorem 2.2 of [3], $f_{p u}^{- 1} (G) \in ω_{s} (X, τ, A)$ . Hence, f_pu is soft ω_s-continuous.

Theorem 3.3

Soft ω_s-continuous functions are soft semi-continuous.

Proof

Let f_pu : (X, τ, A) → (Y, σ, B) be soft ω_s-continuous and let G ∈ σ. By soft ω_s-continuity of f_pu, $f_{p u}^{- 1} (G) \in ω_{s} (X, τ, A)$ . Thus, by Theorem 2.2 of [3], $f_{p u}^{- 1} (G) \in S O (X, τ, A)$ . Hence, f_pu is soft semi-continuous.

The following example shows that the converses of Theorems 3.2 and 3.3 are not true in general:

Example 3.4

Let X = ℝ, Y = {a, b}, A = B = ℤ, τ = {0_A, 1_A, C_ℕ, C_ℚ^c, C_(ℕ∪ℚ^c)}, and σ = {0_B, 1_B, C_{_a_}, C_{_b_}}. It is not difficult to see that Cl_τ_{_ω} (C_ℕ) = C_ℕ, Cl_τ (C_ℕ) = C_ℚ, and Cl_τ_{_ω} (C_{ℚ_c}) = C_ℝ−ℕ. Thus, C_ℚ ∈ SO(X, τ, A) − ω_s(X, τ, A) and C_ℝ−ℕ ∈ ω_s(X, τ, A)−τ. Let p, q : X → Y and u, v : A → B be defined by

\begin{array}{l} p (x) = {\begin{array}{l} a, & if x \in ℕ, \\ b, & if x \in ℝ - ℕ, \end{array} \\ q (x) = {\begin{array}{l} a, & if x \in ℚ^{c}, \\ b, & if x \in ℚ, \end{array} \\ u (a) = v (a) = a for all a \in A . \end{array}

Because $f_{p u}^{- 1} (C_{{a}}) = C_{ℕ} \in τ \subseteq ω_{s} (X, τ, A)$ and $f_{p u}^{- 1} (C_{{b}}) = C_{ℝ - ℕ} \in ω_{s} (X, τ, A) - τ$ , then f_pu is soft ω_s-continuous but not soft continuous. Additionally, because $f_{q v}^{- 1} (C_{{a}}) = C_{ℚ^{c}} \in τ \subseteq S O (X, τ, A)$ and $f_{q v}^{- 1} (C_{{b}}) = C_{ℚ} \in S O (X, τ, A) - ω_{s} (X, τ, A)$ , f_qv is soft semi-continuous but not soft ω_s-continuous.

Theorem 3.5

If f_pu : (X, τ, A) → (Y, σ, B) is soft ω_s-continuous such that (X, τ, A) is soft locally countable, then f_pu is soft continuous.

Proof

Let G ∈ σ. By the soft ω_s-continuity of f_pu, $f_{p u}^{- 1} (G) \in ω_{s} (X, τ, A)$ . Thus, by Theorem 2.8 of [3], $f_{p u}^{- 1} (G) \in τ$ . Hence, f_pu is soft continuous.

Theorem 3.6

If f_pu : (X, τ, A) → (Y, σ, B) is soft semi-continuous such that (X, τ, A) is soft anti-locally countable, then f_pu is soft ω_s-continuous.

Proof

Let G ∈ σ. By the soft semi-continuity of f_pu, $f_{p u}^{- 1} (G) \in S O (X, τ, A)$ . Thus, by Theorem 2.6 of [3], $f_{p u}^{- 1} (G) \in ω_{s} (X, τ, A)$ . Hence, f_pu is soft ω_s-continuous.

Theorem 3.7

For a soft function f_pu : (X, τ, A) → (Y, σ, B), the following are equivalent:

(a) f_pu is soft ω_s-continuous.

(b) For each a_x ∈ SP(X, A) and each G ∈ σ such that f_pu(a_x)∊̃G, there exists H ∈ ω_s (X, τ, A) such that a_x∊̃H and f_pu(H)⊂̃G.

Proof

(a) ⇒ (b): Assume that f_pu is soft ω_s-continuous. Let G ∈ σ such that f_pu(a_x)∊̃G. Because f_pu is soft ω_s-continuous, $f_{p u}^{- 1} (G) \in ω_{s} (X, τ, A)$ . Take $H = f_{p u}^{- 1} (G)$ . Then H ∈ ω_s (X, τ, A) such that a_x∊̃H and f_pu(H)⊂̃G.

(b) ⇒ (a): Let G ∈ σ. For each $a_{x} \tilde{\in} f_{p u}^{- 1} (G)$ , we have f_pu(a_x)∊̃G, and by (b), there exists H_a_{_x} ∈ ω_s (X, τ, A) such that a_x∊̃H_a_{_x} and f_pu(H_a_{_x} )⊂̃G, and thus, $a_{x} \tilde{\in} H_{a_{x}} \tilde{\subseteq} f_{p u}^{- 1} (G)$ . Therefore, $f_{p u}^{- 1} (G) = \tilde{\cup} {H_{a_{x}} : a_{x} \tilde{\in} f_{p u}^{- 1} (G)}$ , and hence $f_{p u}^{- 1} (G) \in ω_{s} (X, τ, A)$ .

Theorem 3.8

For a soft function f_pu : (X, τ, A) → (Y, σ, B), the following are equivalent:

(a) f_pu is soft ω_s-continuous.

(b) For a soft base ℳ of (Y, σ, B), $f_{p u}^{- 1} (M) \in ω_{s} (X, τ, A)$ for all M ∈ ℳ.

(d) For each H ∈ SS(X, A),

f_{p u} (ω_{s} - {C l}_{τ} (H)) \tilde{\subseteq} {C l}_{σ} (f_{p u} (H)) .

(e) For each K ∈ SS(Y, B),

ω_{s} - {C l}_{τ} (f_{p u}^{- 1} (K)) \tilde{\subseteq} f_{p u}^{- 1} ({C l}_{σ} (K)) .

(f) For each K ∈ SS(Y, B),

f_{p u}^{- 1} ({i n t}_{σ} (K)) \tilde{\subseteq} ω_{s} - {i n t}_{τ} (f_{p u}^{- 1} (K)) .

Proof

(a) ⇒ (b): It is obvious.

(b) ⇒(c): Let N ∈ σ^c. Then 1_B – N ∈ σ – {0_B}. Choose ℳ₁ ⊆ ℳ such that 1_B – N = ∪̃{M : B ∈ ℳ₁}. Then

\begin{array}{l} 1_{A} - f_{p u}^{- 1} (N) = f_{p u}^{- 1} (1_{B} - N) \\ = f_{p u}^{- 1} (\tilde{\cup} {M : B \in ℳ_{1}}) \\ = \tilde{\cup} {f_{p u}^{- 1} (M) : M \in ℳ_{1}} . \end{array}

By (b), $f_{p u}^{- 1} (M) \in ω_{s} (X, τ, A)$ for all M ∈ ℳ. Thus, $1_{A} - f_{p u}^{- 1} (N) \in ω_{s} (X, τ, A)$ , and hence $f_{p u}^{- 1} (N)$ is soft ω_s-closed.

(c) ⇒ (d): Let H ∈ SS(X, A). Then Cl_σ (f_pu (H)) ∈ σ^c. So, by (c) $f_{p u}^{- 1} ({C l}_{σ} (f_{p u} (H)))$ is soft ω_s-closed. Because $H \tilde{\subseteq} f_{p u}^{- 1} (f_{p u} (H)) \tilde{\subseteq} f_{p u}^{- 1} ({C l}_{σ} (f_{p u} (H)))$ and $f_{p u}^{- 1} ({C l}_{σ} (f_{p u} (H)))$ is soft ω_s-closed,

ω_{s} - {C l}_{τ} (H) \tilde{\subseteq} f_{p u}^{- 1} ({C l}_{σ} (f_{p u} (H))) .

Thus,

\begin{array}{l} f_{p u} (ω_{s} - {C l}_{τ} (H)) \tilde{\subseteq} f_{p u} (f_{p u}^{- 1} ({C l}_{σ} (f_{p u} (H)))) \\ \tilde{\subseteq} {C l}_{σ} (f_{p u} (H))) . \end{array}

(d) ⇒ (e): Let K ∈ SS(Y, B). Then by (d),

\begin{array}{l} f_{p u} (ω_{s} - {C l}_{τ} (f_{p u}^{- 1} (K))) \tilde{\subseteq} {C l}_{σ} (f_{p u} (f_{p u}^{- 1} (K))) \\ \tilde{\subseteq} {C l}_{σ} (K) . \end{array}

Therefore,

\begin{array}{l} ω_{s} - {C l}_{τ} (f_{p u}^{- 1} (K)) \tilde{\subseteq} f_{p u}^{- 1} (f_{p u} (ω_{s} - {C l}_{τ} (f_{p u}^{- 1} (K)))) \\ \tilde{\subseteq} f_{p u}^{- 1} ({C l}_{σ} (K)) . \end{array}

(e) ⇒ (f): Let K ∈ SS(Y, B). Then by (e), $ω_{s} - {C l}_{τ} (f_{p u}^{- 1} (1_{B} - K)) \tilde{\subseteq} f_{p u}^{- 1} ({C l}_{σ} (1_{B} - K))$ . In addition, by Theorem 3.33(c) of [3],

\begin{array}{l} 1_{A} - ω_{s} - {C l}_{τ} (1_{A} - f_{p u}^{- 1} (K)) \\ = ω_{s} - {i n t}_{τ} (f_{p u}^{- 1} (K)) . \end{array}

Thus,

\begin{array}{l} f_{p u}^{- 1} ({i n t}_{σ} (K))) = f_{p u}^{- 1} (1_{B} - {C l}_{σ} (1_{B} - K)) \\ = 1_{A} - f_{p u}^{- 1} ({C l}_{σ} (1_{B} - K)) \\ \tilde{\subseteq} 1_{A} - ω_{s} - {C l}_{τ} (f_{p u}^{- 1} (1_{B} - K)) \\ = 1_{A} - ω_{s} - {C l}_{τ} (1_{A} - f_{p u}^{- 1} (K)) \\ = ω_{s} - {i n t}_{τ} (f_{p u}^{- 1} (K)) . \end{array}

(f) ⇒ (a): Let G ∈ σ. Then G = int_σ (G). Thus, by (f), $f_{p u}^{- 1} (G) \tilde{\subseteq} ω_{s} - {i n t}_{τ} (f_{p u}^{- 1} (G))$ . Therefore, $f_{p u}^{- 1} (G) = ω_{s} - {i n t}_{τ} (f_{p u}^{- 1} (G))$ and hence $f_{p u}^{- 1} (G) \in f_{p u}^{- 1} (G) \in ω_{s} (X, τ, A)$ .

Lemma 3.9

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

ω_{s} - {C l}_{τ} (H) = H \tilde{\cup} {i n t}_{τ_{ω}} ({C l}_{τ} (H)) .

Proof

Because ω_s–Cl_τ (H) is soft ω_s-closed, then by Theorem 2.21 of [3], int_τ_{_ω} (Cl_τ (ω_s-Cl_τ (H))) ⊂̃ ω_s-Cl_τ (H). Thus, int_τ_{_ω}(Cl_τ (H)) ⊂̃ int_τ_{_ω}(Cl_τ (ω_s-Cl_τ (H))) ⊂̃ ω_s-Cl_τ (H). Hence, H ∪̃ int_τ_{_ω} (Cl_τ (H)) ⊂̃ ω_s-Cl_τ (H). To show that ω_s-Cl_τ (H) ⊂̃ H ∪̃ int_τ_{_ω} (Cl_τ (H)), it is sufficient to show that H ∪̃ int_τ_{_ω} (Cl_τ (H)) is soft ω_s-closed. Because int_τ_{_ω} (Cl_τ (H)) ⊂̃ Cl_τ (H),

{C l}_{τ} ({i n t}_{τ_{ω}} ({C l}_{τ} (H))) \tilde{\subseteq} {C l}_{τ} (H) .

Thus,

\begin{array}{l} {i n t}_{τ_{ω}} ({C l}_{τ} (H \tilde{\cup} {i n t}_{τ_{ω}} ({C l}_{τ} (H)))) \\ = {i n t}_{τ_{ω}} ({C l}_{τ} (H) \tilde{\cup} {C l}_{τ} ({i n t}_{τ_{ω}} ({C l}_{τ} (H)))) \\ = {i n t}_{τ_{ω}} ({C l}_{τ} (H)) \\ \tilde{\subseteq} H \tilde{\cup} {i n t}_{τ_{ω}} ({C l}_{τ} (H)) . \end{array}

Therefore, by Theorem 2.21 of [3], H ∪̃ int_τ_{_ω} (Cl_τ (H)) is soft ω_s-closed.

Theorem 3.10

For a soft function f_pu : (X, τ, A) → (Y, σ, B), the following are equivalent:

(a) f_pu is soft ω_s-continuous.

(b) For each H ∈ SS(X, A),

f_{p u} ({i n t}_{τ_{ω}} ({C l}_{τ} (H)) \tilde{\subseteq} {C l}_{σ} (f_{p u} (H)) .

{i n t}_{τ_{ω}} ({C l}_{τ} (f_{p u}^{- 1} (K))) \subseteq f_{p u}^{- 1} ({C l}_{σ} (H)) .

Proof

(a) ⇒ (b): Suppose that f_pu is soft ω_s-continuous. Let H ∈ SS(X, A). Then by part (d) of Theorem 3.8,

f_pu (ω_s-Cl_τ (H)) ⊂̃ Cl_σ (f_pu (H)). Thus, by Lemma 3.9, it follows that

f_{p u} ({i n t}_{τ_{ω}} ({C l}_{τ} (H)) \tilde{\subseteq} f_{p u} (ω_{s} - {C l}_{τ} (H)) \tilde{\subseteq} {C l}_{σ} (f_{p u} (H)) .

(b) ⇒(a): We apply Theorem 3.8(d). Let H ∈ SS(X, A). Then by (b),

f_{p u} ({i n t}_{τ_{ω}} ({C l}_{τ} (H)) \tilde{\subseteq} {C l}_{σ} (f_{p u} (H)) .

Thus, by Lemma 3.9,

\begin{array}{l} f_{p u} (ω_{s} - {C l}_{τ} (H)) = f_{p u} (H \tilde{\cup} {i n t}_{τ_{ω}} ({C l}_{τ} (H))) \\ = f_{p u} (H) \tilde{\cup} f_{p u} ({i n t}_{τ_{ω}} ({C l}_{τ} (H))) \\ \tilde{\subseteq} {C l}_{σ} (f_{p u} (H)) . \end{array}

(a) ⇒ (c): Suppose that f_pu is soft ω_s-continuous and let K ∈ SS(Y, B). Then by Theorem 3.8(e), $ω_{s} - {C l}_{τ} (f_{p u}^{- 1} (K)) \tilde{\subseteq} f_{p u}^{- 1} ({C l}_{σ} (K))$ . Therefore, by Lemma 3.9, it follows that

\begin{array}{l} {i n t}_{τ_{ω}} ({C l}_{τ} (f_{p u}^{- 1} (K))) \tilde{\subseteq} ω_{s} - {C l}_{τ} (f_{p u}^{- 1} (K)) \\ \tilde{\subseteq} f_{p u}^{- 1} ({C l}_{σ} (K)) . \end{array}

(c) ⇒ (a): We apply Theorem 3.8(e). Let K ∈ SS(Y, B). Then by (c), we have ${i n t}_{τ_{ω}} ({C l}_{τ} (f_{p u}^{- 1} (K))) \tilde{\subseteq} f_{p u}^{- 1} ({C l}_{σ} (K))$ . Thus, by Lemma 3.9, it follows that

\begin{array}{l} ω_{s} - {C l}_{τ} (f_{p u}^{- 1} (K)) = f_{p u}^{- 1} (K) \tilde{\cup} {i n t}_{τ_{ω}} ({C l}_{τ} (f_{p u}^{- 1} (K))) \\ \tilde{\subseteq} f_{p u}^{- 1} ({C l}_{σ} (K)) . \end{array}

Theorem 3.11

If f_pu : (X, τ, A) → (Y, σ, B) is soft ω_s-continuous and if f_qv : (Y, σ, B) → (Z, δ, C) is soft continuous, then f₍_q_∘_p₎₍_v_∘_u₎ : (X, τ, A) → (Z, δ, C) is soft ω_s-continuous.

Proof

Let K ∈ δ. Because f_qv : (Y, σ,, B) → (Z, δ, C) is soft continuous, $f_{q v}^{- 1} (K) \in σ$ . Because f_pu is soft ω_s-continuous, $f_{p u}^{- 1} (f_{q v}^{- 1} (K)) \in ω_{s} (X, τ, A)$ . However, $f_{p u}^{- 1} (f_{q v}^{- 1} (K)) = f_{(q \circ p) (v \circ u)}^{- 1} (K)$ . Therefore, $f_{(q \circ p) (v \circ u)}^{- 1} (K) \in ω_{s} (X, τ, A)$ . Hence, f₍_q_∘_p₎₍_v_∘_u₎ : (X, τ, A) → (Z, δ, C) is soft ω_s-continuous.

The composition of two soft ω_s-continuous functions is not required to be soft ω_s-continuous, in general:

Example 3.12

Let ℑ be the usual topology on ℝ and let τ = {F ∈ SS(ℝ, ℕ) : F (a) ∈ ℑ for all a ∈ ℕ}. Define p, q :

ℝ → ℝ and u, v : ℕ → ℕ as follows:

\begin{array}{l} p (x) = {\begin{array}{l} x, & if x \leq 1, \\ 0, & if x > 1, \end{array} \\ q (x) = {\begin{array}{l} 0, & if x < 1, \\ 3, & if x \geq 1, \end{array} \\ u (a) = v (a) = a for all a \in ℕ . \end{array}

Then

(q \circ p) (x) = {\begin{array}{l} 0, & if x \neq 1, \\ 3, & if x = 1, \end{array}

and

(v \circ u) (a) = a for all a \in ℕ .

f_pu and f_qv are obviously soft semi-continuous, and (ℝ, τ_u, ℕ) is soft anti-locally countable; thus, by Theorem 3.6, f_pu and f_qv are soft ω_s-continuous. Let G ∈ SS(ℝ, ℕ), where G = C_{(2, ∞)}. Then G ∈ τ, but ${(f_{q v} \circ f_{p u})}^{- 1} (G) = (f_{(q \circ p) (v \circ u)}^{- 1}) (G) = H$ , where H(a) = {1} for all a ∈ ℕ. Thus, (f_qv ∘ f_pu)⁻¹ (G) ∉ ω_s(ℝ, τ, ℕ). Hence, f_qv ∘ f_pu is not soft ω_s-continuous.

Lemma 3.13

Let (X, ℑ) be a TS and let A be a set of parameters. Let H ∈ SS(X, A). Then H ∈ ω_s(X, τ (ℑ), A) if and only if H (a) ∈ ω_s(X, ℑ) for all a ∈ A.

Proof

For each a ∈ A, set ℑ_a = ℑ. Then $τ (I) = \underset{a \in A}{\oplus} I_{a}$ . Then by Theorem 4.8 of [3], we obtain the result.

Theorem 3.14

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 p : (X, ℑ) → (Y, ℵ) is ω_s-continuous if and only if f_pu : (X, τ (ℑ), A) → (Y, τ (ℵ), B) is soft ω_s-continuous.

Proof

Necessity

Suppose that p : (X, ℑ) → (Y, ℵ) is ω_s-continuous. Let G ∈ τ (ℵ). Then for each a ∈ A, G(u (a)) ∈ ℵ, and so p⁻¹(G(u (a))) ∈ ω_s(X, ℑ). Therefore, $(f_{p u}^{- 1} (G)) (a) = p^{- 1} (G (u (a))) \in ω_{s} (X, I)$ for all a ∈ A. Hence, by Lemma 3.13, $f_{p u}^{- 1} (G) \in ω_{s} (X, τ (I), A)$ .

Sufficiency

Suppose that f_pu : (X, τ (ℑ), A) → (Y, τ (ℵ), B) is soft ω_p-continuous. Let V ∈ ℵ. Then C_V ∈ τ (ℵ), and so $f_{p u}^{- 1} (C_{V}) \in ω_{s} (X, τ (I), A)$ . Pick a ∈ A. Then, by Lemma 3.13, $(f_{p u}^{- 1} (C_{V})) (a) = p^{- 1} (V) \in ω_{s} (X, I)$ .

4. Conclusion

Go to

Abstract
1. Introduction and Preliminaries
2. Soft ω-Continuity
3. Soft ω_s-Continuity
4. Conclusion
Conflict of Interest
References
Biography

The classes of ω-continuous functions and soft ω_s-continuous functions were extended to include STSs. Also, soft. Several characterizations, relationships, and examples were given. The following topics could be considered in future studies: defining soft ω-open functions and defining soft ω_s-open functions.

Conflict of Interest

Go to

Abstract
1. Introduction and Preliminaries
2. Soft ω-Continuity
3. Soft ω_s-Continuity
4. Conclusion
Conflict of Interest
References
Biography

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

References

Go to

Abstract
1. Introduction and Preliminaries
2. Soft ω-Continuity
3. Soft ω_s-Continuity
4. Conclusion
Conflict of Interest
References
Biography

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Biography

Go to

Abstract
1. Introduction and Preliminaries
2. Soft ω-Continuity
3. Soft ω_s-Continuity
4. Conclusion
Conflict of Interest
References
Biography

Samer Al Ghour received his Ph.D. in Mathematics from the University of Jordan, Jordan in 1999. Currently, he is a professor in 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(2): 183-192

Published online June 25, 2022 https://doi.org/10.5391/IJFIS.2022.22.2.183

Soft -Continuity and Soft -Continuity in Soft Topological Spaces

Samer Al Ghour

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

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

Received: November 28, 2021; Revised: January 16, 2022; Accepted: January 26, 2022

Abstract

Keywords: Soft semi-continuous functions, Soft ω-open sets, ω-continuity, ω_s-continuity, Soft generated soft topological space, Soft induced topological spaces

1. Introduction and Preliminaries

The following definitions and results are used in the following:

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

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

(c) $G (a) = {\begin{array}{l} {x}, & if a = e, \\ \emptyset, & if a \neq e \end{array}$ , is denoted by e_x, and is called a soft point.

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

Definition 1.2 [31]

LetG ∈ SS(X, A) and e_x ∈ SP(X, A). Then e_x is said to belong to F (notation: e_x ∊̃ G) if e_x ∊̃ G or, equivalently, e_x ∊̃ G if and only if x ∈ G(e).

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 [32]

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

An STS (X, τ, A) is called

(a) [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.

(b) [2] soft anti-locally countable if for every F ∈ τ −{0_A}, F is not a countable soft set.

Definition 1.7

Let (X,ℑ) be a ST, and let D ⊆ X. D is called

(a) [33] semi-open if there exists V ∈ ℑ such that V ⊆ D ⊆ Cl_ℑ (V ). SO(X, ℑ) denotes the family of all semi-open sets in (X,ℑ).

(b) [34] ω_s-open if there exists V ∈ ℑ such that V ⊆ D ⊆ Cl_ℑ_ω (V ). ω_s(X, ℑ) denotes the family of all ω_s-open sets in (X, ℑ).

Similar types of generalized sets in ideal STs have been discussed in [35–39].

Definition 1.8 [40]

Definition 1.9 [33]

A function g : (X, ℑ) → (Y, ℵ) is called semi-continuous if for each V ∈ ℵ, g⁻¹(V ) ∈ SO(X, ℑ).

Definition 1.10 [34]

A function g : (X, ℑ) → (Y, ℵ) is called ω_s-continuous if for each V ∈ ℵ, g⁻¹(V ) ∈ ω_s(X, ℑ).

Definition 1.11

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

(a) [30] K is called a soft semi-open set in (X, τ, A) if there exists F ∈ τ such that F ⊂̃ K ⊂̃ Cl_τ (F). SO(X, τ, A) denotes the family of all soft semi-open sets in (X, τ, A).

(b) [30] K is called a soft semi-closed set in (X, τ, A) if 1_A − K ∈ SO(X, τ, A).

(c) [3] K is called a soft ω_s-open set in (X, τ, A) if there exists F ∈ τ such that F ⊂̃ K ⊂̃ Cl_τω (F). ω_s(X, τ, A) denotes the family of all soft ω_s-open sets in (X, τ, A).

Definition 1.12 [42]

A soft function f_pu : (X, τ, A) → (Y, σ, B) is called soft semi-continuous if for every G ∈ σ, $f_{p u}^{- 1} (G) \in S O (X, τ, A)$ .

Definition 1.13 [3]

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

(a) The soft ω_s-closure of H in (X, τ, A) is denoted by ω_s-Cl_τ (H) and defined by

ω_{s} - {C l}_{τ} (H) = \tilde{\cap} {M : M is soft ω_{s} -closed in (X, τ, A) and H \tilde{\subseteq} M} .

(b) The soft ω_s-interior of H in (X, τ,A) is denoted by ω_sint_τ (H) and defined by

ω_{s} - {i n t}_{τ} (H) = \tilde{\cup} {K : K is soft ω_{s} -open in (X, τ, A) and K \tilde{\subseteq} H} .

2. Soft ω-Continuity

Definition 2.1

Theorem 2.2

For a soft function f_pu : (X, τ,A) → (Y, σ,B), the following conditions are equivalent:

(a) f_pu is soft ω-continuous.

(b) For each G ∈ σ, $f_{p u}^{- 1} (G) \in τ_{ω}$ .

(d) For a soft subbase of (Y, σ,B), $f_{p u}^{- 1} (S) \in τ_{ω}$ for all .

(e) For each N ∈ σ^c, $f_{p u}^{- 1} (N) \in {(τ_{ω})}^{c}$ .

(f) For each K ∈ SS(X,A), f_pu(Cl_τ_{_ω} (K))⊂̃Cl_σ (f_pu(K)).

(g) For each L ∈ SS(Y,B), ${C l}_{τ_{ω}} (f_{p u}^{- 1} (L)) \tilde{\subseteq} f_{p u}^{- 1} ({C l}_{σ} (L))$ .

Proof

(b) =⇒ (c): Obvious.

f_{p u} (a_{x}) \tilde{\in} S_{1} \tilde{\cap} S_{2} \tilde{\cap} \dots \tilde{\cap} S_{n} \tilde{\subseteq} 1_{B} - N .

So,

\begin{array}{l} a_{x} \tilde{\in} f_{p u}^{- 1} (S_{1}) \tilde{\cap} f_{p u}^{- 1} (S_{2}) \tilde{\cap} \dots \tilde{\cap} f_{p u}^{- 1} (S_{n}) \\ \tilde{\subseteq} f_{p u}^{- 1} (1_{B} - N) = 1_{A} - f_{p u}^{- 1} (N) . \end{array}

By (d), $f_{p u}^{- 1} (S_{1}) \tilde{\cap} f_{p u}^{- 1} (S_{2}) \tilde{\cap} \dots \tilde{\cap} f_{p u}^{- 1} (S_{n}) \in τ_{ω}$ , and thus, $1_{A} - f_{p u}^{- 1} (N) \in τ_{ω}$ .

{C l}_{τ_{ω}} (K) \tilde{\subseteq} f_{p u}^{- 1} ({C l}_{σ} (f_{p u} (K))),

and so,

\begin{array}{l} f_{p u} ({C l}_{τ_{ω}} (K)) \tilde{\subseteq} f_{p u} (f_{p u}^{- 1} ({C l}_{σ} (f_{p u} (K)))) \\ \tilde{\subseteq} {C l}_{σ} (f_{p u} (K)) . \end{array}

(f) ⇒ (g): Let L ∈ SS(Y,B). Then $f_{p u}^{- 1} (L) \in S S (X, A)$ . Thus, by (f),

f_{p u} ({C l}_{τ_{ω}} (f_{p u}^{- 1} (L))) \tilde{\subseteq} {C l}_{σ} (f_{p u} (f_{p u}^{- 1} (L))) \tilde{\subseteq} {C l}_{σ} (L) .

Therefore,

\begin{array}{l} {C l}_{τ_{ω}} (f_{p u}^{- 1} (L)) \tilde{\subseteq} f_{p u}^{- 1} (f_{p u} ({C l}_{τ_{ω}} (f_{p u}^{- 1} (L)))) \\ \tilde{\subseteq} f_{p u}^{- 1} ({C l}_{σ} (L)) . \end{array}

Theorem 2.3

If f_pu : (X, τ,A) → (Y, σ,B) is soft ω-continuous at a_x ∈ SP(X,A), then p : (X, τ_a) → (Y, σ_u₍_a₎) is ω-continuous at x.

Proof

Theorem 2.4

If f_pu : (X, τ,A) → (Y, σ,B) is soft ω-continuous, then p : (X, τ_a) → (Y, σ_u₍_a₎) is ω-continuous for all a ∈ A.

Proof

Theorem 2.5

Proof

Necessity

Sufficiency

Suppose that f_pu : (X, τ(ℑ),A) → (Y, τ(ℵ), B) is soft ω_p-continuous at a_x. Then by Theorem 2.3,

p : (X, (τ (ℑ))_a) → (Y, (τ (ℵ))_u₍_a₎) is ω-continuous at x. Because (τ (ℑ))_a = ℑ and (τ (ℵ))_u₍_a₎ = ℵ, p : (X, ℑ) → (Y, ℵ) is ω-continuous at x.

Corollary 2.6

Theorem 2.7

Soft continuous soft functions are soft ω-continuous.

Proof

Follows from Theorem 2.2 and Theorem 2(a) of [2].

The following example shows that the converse of Theorem 2.7 is not true in general:

Example 2.8

Theorem 2.9

If f_pu : (X, τ,A) → (Y, σ,B) is soft ω-continuous and if f_qv : (Y, σ,B) → (Z, δ,C) is soft continuous, then f₍_q_◦_p₎₍_v_◦_u₎ : (X, τ,A) → (Z, δ,C) is soft ω-continuous.

Proof

Theorem 2.10

If f_pu : (X, τ,A) → (Y, σ,B) is soft ω-continuous and Z is a non-empty subset of X, then the soft restriction f(_p_{_{|_Z}})_u : (Z, τ_Z,A) → (Y, σ,B) is soft ω-continuous.

Proof

Theorem 2.11

Proof

We apply statement (e) of Theorem 2.2. Let N ∈ σ^c. Then

\begin{array}{l} f_{p u}^{- 1} (N) = f_{p u}^{- 1} (N) \tilde{\cap} 1_{A} \\ = f_{p u}^{- 1} (N) \tilde{\cap} (C_{Z} \tilde{\cup} C_{W}) \\ = (f_{p u}^{- 1} (N) \tilde{\cap} C_{Z}) \tilde{\cup} (f_{p u}^{- 1} (N) \tilde{\cap} C_{W}) \\ = (f_{(p_{∣ Z}) u}^{- 1} (N)) \tilde{\cup} (f_{(p_{∣ W}) u}^{- 1} (N)) . \end{array}

Because f(_p_{_{|_Z}})_u and f(_p_{_{|_W}})_u are soft ω-continuous, then

f_{(p_{∣ Z .}) u}^{- 1} (N) \in {({(τ_{Z})}_{ω})}^{c} and f_{(p_{∣ W .}) u}^{- 1} (N) \in {({(τ_{W})}_{ω})}^{c} .

Theorem 2.12

Proof

Corollary 2.13

Proof

3. Soft ω_s-Continuity

Definition 3.1

A soft function f_pu : (X, τ, A) → (Y, σ, B) is called soft ω_s-continuous if for every G ∈ σ, $f_{p u}^{- 1} (G) \in ω_{s} (X, τ, A)$ .

Theorem 3.2

Soft continuous functions are soft ω_s-continuous.

Proof

Theorem 3.3

Soft ω_s-continuous functions are soft semi-continuous.

Proof

The following example shows that the converses of Theorems 3.2 and 3.3 are not true in general:

Example 3.4

\begin{array}{l} p (x) = {\begin{array}{l} a, & if x \in ℕ, \\ b, & if x \in ℝ - ℕ, \end{array} \\ q (x) = {\begin{array}{l} a, & if x \in ℚ^{c}, \\ b, & if x \in ℚ, \end{array} \\ u (a) = v (a) = a for all a \in A . \end{array}

Theorem 3.5

If f_pu : (X, τ, A) → (Y, σ, B) is soft ω_s-continuous such that (X, τ, A) is soft locally countable, then f_pu is soft continuous.

Proof

Let G ∈ σ. By the soft ω_s-continuity of f_pu, $f_{p u}^{- 1} (G) \in ω_{s} (X, τ, A)$ . Thus, by Theorem 2.8 of [3], $f_{p u}^{- 1} (G) \in τ$ . Hence, f_pu is soft continuous.

Theorem 3.6

If f_pu : (X, τ, A) → (Y, σ, B) is soft semi-continuous such that (X, τ, A) is soft anti-locally countable, then f_pu is soft ω_s-continuous.

Proof

Theorem 3.7

For a soft function f_pu : (X, τ, A) → (Y, σ, B), the following are equivalent:

(a) f_pu is soft ω_s-continuous.

(b) For each a_x ∈ SP(X, A) and each G ∈ σ such that f_pu(a_x)∊̃G, there exists H ∈ ω_s (X, τ, A) such that a_x∊̃H and f_pu(H)⊂̃G.

Proof

Theorem 3.8

For a soft function f_pu : (X, τ, A) → (Y, σ, B), the following are equivalent:

(a) f_pu is soft ω_s-continuous.

(b) For a soft base ℳ of (Y, σ, B), $f_{p u}^{- 1} (M) \in ω_{s} (X, τ, A)$ for all M ∈ ℳ.

(d) For each H ∈ SS(X, A),

f_{p u} (ω_{s} - {C l}_{τ} (H)) \tilde{\subseteq} {C l}_{σ} (f_{p u} (H)) .

(e) For each K ∈ SS(Y, B),

ω_{s} - {C l}_{τ} (f_{p u}^{- 1} (K)) \tilde{\subseteq} f_{p u}^{- 1} ({C l}_{σ} (K)) .

(f) For each K ∈ SS(Y, B),

f_{p u}^{- 1} ({i n t}_{σ} (K)) \tilde{\subseteq} ω_{s} - {i n t}_{τ} (f_{p u}^{- 1} (K)) .

Proof

(a) ⇒ (b): It is obvious.

(b) ⇒(c): Let N ∈ σ^c. Then 1_B – N ∈ σ – {0_B}. Choose ℳ₁ ⊆ ℳ such that 1_B – N = ∪̃{M : B ∈ ℳ₁}. Then

\begin{array}{l} 1_{A} - f_{p u}^{- 1} (N) = f_{p u}^{- 1} (1_{B} - N) \\ = f_{p u}^{- 1} (\tilde{\cup} {M : B \in ℳ_{1}}) \\ = \tilde{\cup} {f_{p u}^{- 1} (M) : M \in ℳ_{1}} . \end{array}

By (b), $f_{p u}^{- 1} (M) \in ω_{s} (X, τ, A)$ for all M ∈ ℳ. Thus, $1_{A} - f_{p u}^{- 1} (N) \in ω_{s} (X, τ, A)$ , and hence $f_{p u}^{- 1} (N)$ is soft ω_s-closed.

ω_{s} - {C l}_{τ} (H) \tilde{\subseteq} f_{p u}^{- 1} ({C l}_{σ} (f_{p u} (H))) .

Thus,

\begin{array}{l} f_{p u} (ω_{s} - {C l}_{τ} (H)) \tilde{\subseteq} f_{p u} (f_{p u}^{- 1} ({C l}_{σ} (f_{p u} (H)))) \\ \tilde{\subseteq} {C l}_{σ} (f_{p u} (H))) . \end{array}

(d) ⇒ (e): Let K ∈ SS(Y, B). Then by (d),

\begin{array}{l} f_{p u} (ω_{s} - {C l}_{τ} (f_{p u}^{- 1} (K))) \tilde{\subseteq} {C l}_{σ} (f_{p u} (f_{p u}^{- 1} (K))) \\ \tilde{\subseteq} {C l}_{σ} (K) . \end{array}

Therefore,

\begin{array}{l} ω_{s} - {C l}_{τ} (f_{p u}^{- 1} (K)) \tilde{\subseteq} f_{p u}^{- 1} (f_{p u} (ω_{s} - {C l}_{τ} (f_{p u}^{- 1} (K)))) \\ \tilde{\subseteq} f_{p u}^{- 1} ({C l}_{σ} (K)) . \end{array}

(e) ⇒ (f): Let K ∈ SS(Y, B). Then by (e), $ω_{s} - {C l}_{τ} (f_{p u}^{- 1} (1_{B} - K)) \tilde{\subseteq} f_{p u}^{- 1} ({C l}_{σ} (1_{B} - K))$ . In addition, by Theorem 3.33(c) of [3],

\begin{array}{l} 1_{A} - ω_{s} - {C l}_{τ} (1_{A} - f_{p u}^{- 1} (K)) \\ = ω_{s} - {i n t}_{τ} (f_{p u}^{- 1} (K)) . \end{array}

Thus,

\begin{array}{l} f_{p u}^{- 1} ({i n t}_{σ} (K))) = f_{p u}^{- 1} (1_{B} - {C l}_{σ} (1_{B} - K)) \\ = 1_{A} - f_{p u}^{- 1} ({C l}_{σ} (1_{B} - K)) \\ \tilde{\subseteq} 1_{A} - ω_{s} - {C l}_{τ} (f_{p u}^{- 1} (1_{B} - K)) \\ = 1_{A} - ω_{s} - {C l}_{τ} (1_{A} - f_{p u}^{- 1} (K)) \\ = ω_{s} - {i n t}_{τ} (f_{p u}^{- 1} (K)) . \end{array}

Lemma 3.9

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

ω_{s} - {C l}_{τ} (H) = H \tilde{\cup} {i n t}_{τ_{ω}} ({C l}_{τ} (H)) .

Proof

{C l}_{τ} ({i n t}_{τ_{ω}} ({C l}_{τ} (H))) \tilde{\subseteq} {C l}_{τ} (H) .

Thus,

\begin{array}{l} {i n t}_{τ_{ω}} ({C l}_{τ} (H \tilde{\cup} {i n t}_{τ_{ω}} ({C l}_{τ} (H)))) \\ = {i n t}_{τ_{ω}} ({C l}_{τ} (H) \tilde{\cup} {C l}_{τ} ({i n t}_{τ_{ω}} ({C l}_{τ} (H)))) \\ = {i n t}_{τ_{ω}} ({C l}_{τ} (H)) \\ \tilde{\subseteq} H \tilde{\cup} {i n t}_{τ_{ω}} ({C l}_{τ} (H)) . \end{array}

Therefore, by Theorem 2.21 of [3], H ∪̃ int_τ_{_ω} (Cl_τ (H)) is soft ω_s-closed.

Theorem 3.10

For a soft function f_pu : (X, τ, A) → (Y, σ, B), the following are equivalent:

(a) f_pu is soft ω_s-continuous.

(b) For each H ∈ SS(X, A),

f_{p u} ({i n t}_{τ_{ω}} ({C l}_{τ} (H)) \tilde{\subseteq} {C l}_{σ} (f_{p u} (H)) .

{i n t}_{τ_{ω}} ({C l}_{τ} (f_{p u}^{- 1} (K))) \subseteq f_{p u}^{- 1} ({C l}_{σ} (H)) .

Proof

(a) ⇒ (b): Suppose that f_pu is soft ω_s-continuous. Let H ∈ SS(X, A). Then by part (d) of Theorem 3.8,

f_pu (ω_s-Cl_τ (H)) ⊂̃ Cl_σ (f_pu (H)). Thus, by Lemma 3.9, it follows that

f_{p u} ({i n t}_{τ_{ω}} ({C l}_{τ} (H)) \tilde{\subseteq} f_{p u} (ω_{s} - {C l}_{τ} (H)) \tilde{\subseteq} {C l}_{σ} (f_{p u} (H)) .

(b) ⇒(a): We apply Theorem 3.8(d). Let H ∈ SS(X, A). Then by (b),

f_{p u} ({i n t}_{τ_{ω}} ({C l}_{τ} (H)) \tilde{\subseteq} {C l}_{σ} (f_{p u} (H)) .

Thus, by Lemma 3.9,

\begin{array}{l} f_{p u} (ω_{s} - {C l}_{τ} (H)) = f_{p u} (H \tilde{\cup} {i n t}_{τ_{ω}} ({C l}_{τ} (H))) \\ = f_{p u} (H) \tilde{\cup} f_{p u} ({i n t}_{τ_{ω}} ({C l}_{τ} (H))) \\ \tilde{\subseteq} {C l}_{σ} (f_{p u} (H)) . \end{array}

\begin{array}{l} {i n t}_{τ_{ω}} ({C l}_{τ} (f_{p u}^{- 1} (K))) \tilde{\subseteq} ω_{s} - {C l}_{τ} (f_{p u}^{- 1} (K)) \\ \tilde{\subseteq} f_{p u}^{- 1} ({C l}_{σ} (K)) . \end{array}

\begin{array}{l} ω_{s} - {C l}_{τ} (f_{p u}^{- 1} (K)) = f_{p u}^{- 1} (K) \tilde{\cup} {i n t}_{τ_{ω}} ({C l}_{τ} (f_{p u}^{- 1} (K))) \\ \tilde{\subseteq} f_{p u}^{- 1} ({C l}_{σ} (K)) . \end{array}

Theorem 3.11

Proof

The composition of two soft ω_s-continuous functions is not required to be soft ω_s-continuous, in general:

Example 3.12

Let ℑ be the usual topology on ℝ and let τ = {F ∈ SS(ℝ, ℕ) : F (a) ∈ ℑ for all a ∈ ℕ}. Define p, q :

ℝ → ℝ and u, v : ℕ → ℕ as follows:

\begin{array}{l} p (x) = {\begin{array}{l} x, & if x \leq 1, \\ 0, & if x > 1, \end{array} \\ q (x) = {\begin{array}{l} 0, & if x < 1, \\ 3, & if x \geq 1, \end{array} \\ u (a) = v (a) = a for all a \in ℕ . \end{array}

Then

(q \circ p) (x) = {\begin{array}{l} 0, & if x \neq 1, \\ 3, & if x = 1, \end{array}

and

(v \circ u) (a) = a for all a \in ℕ .

Lemma 3.13

Let (X, ℑ) be a TS and let A be a set of parameters. Let H ∈ SS(X, A). Then H ∈ ω_s(X, τ (ℑ), A) if and only if H (a) ∈ ω_s(X, ℑ) for all a ∈ A.

Proof

For each a ∈ A, set ℑ_a = ℑ. Then $τ (I) = \underset{a \in A}{\oplus} I_{a}$ . Then by Theorem 4.8 of [3], we obtain the result.

Theorem 3.14

Proof

Necessity

Sufficiency

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International Journalof Fuzzy Logic andIntelligent Systems

Original Article

Soft -Continuity and Soft -Continuity in Soft Topological Spaces

Abstract

1. Introduction and Preliminaries

Definition 1.1

Definition 1.2 [31]

Theorem 1.3 [5]

Theorem 1.4 [32]

Theorem 1.5 [1]

Definition 1.6

Definition 1.7

Definition 1.8 [40]

Definition 1.9 [33]

Definition 1.10 [34]

Definition 1.11

Definition 1.12 [42]

Definition 1.13 [3]

2. Soft ω-Continuity

Definition 2.1

Theorem 2.2

Theorem 2.3

Theorem 2.4

Theorem 2.5

Corollary 2.6

Theorem 2.7

Example 2.8

Theorem 2.9

Theorem 2.10

Theorem 2.11

Theorem 2.12

Corollary 2.13

3. Soft ωs-Continuity

Definition 3.1

Theorem 3.2

Theorem 3.3

Example 3.4

Theorem 3.5

Theorem 3.6

Theorem 3.7

Theorem 3.8

Lemma 3.9

Theorem 3.10

Theorem 3.11

Example 3.12

Lemma 3.13

Theorem 3.14

4. Conclusion

Conflict of Interest

References

Biography

Article

Original Article

Soft -Continuity and Soft -Continuity in Soft Topological Spaces

Abstract

1. Introduction and Preliminaries

Definition 1.1

Definition 1.2 [31]

Theorem 1.3 [5]

Theorem 1.4 [32]

Theorem 1.5 [1]

Definition 1.6

Definition 1.7

Definition 1.8 [40]

Definition 1.9 [33]

Definition 1.10 [34]

Definition 1.11

Definition 1.12 [42]

Definition 1.13 [3]

2. Soft ω-Continuity

Definition 2.1

Theorem 2.2

Theorem 2.3

Theorem 2.4

Theorem 2.5

Corollary 2.6

Theorem 2.7

Example 2.8

Theorem 2.9

Theorem 2.10

International Journal
of Fuzzy Logic and
Intelligent Systems

3. Soft ω_s-Continuity

3. Soft ω_s-Continuity