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

© The Korean Institute of Intelligent Systems

Samer Al Ghour

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

**Correspondence to : **

Samer Al Ghour (algore@just.edu.jo)

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.

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

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 _{A}_{A}_{A}_{A}^{c}_{s}_{s}_{s}_{s}_{s}_{s}_{s}

Let (_{τ}_{τ}_{ℑ} (

The following definitions and results are used in the following:

Let

(a) _{Y}

(b) [1] _{Y}

(c) _{x}

The set of all soft points in

Let_{x}_{x}_{x}_{x}_{x}

Let (_{a}

Let (

defines a soft topology on

Let _{a}

Then

An STS (

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

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

Let (

(a) [33] semi-open if there exists _{ℑ} (

(b) [34] _{s}_{ℑ}_{ω}_{s}_{s}

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

A function _{ω}

A function ^{−1}(

A function _{s}^{−1}(_{s}

Let (

(a) [30] _{τ}

(b) [30] _{A}

(c) [3] _{s}_{τω}_{s}_{s}

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].

A soft function _{pu}

Let (

(a) The soft _{s}_{s-}_{τ}

(b) The soft _{s}_{s}_{τ}

In this section, we introduce the concept of soft

A soft function _{pu}_{x}_{pu}_{x}_{ω}_{x}_{pu}_{pu}_{x}_{pu}

For a soft function _{pu}

(a) _{pu}

(b) For each

(c) For a soft base ℳ of (

(d) For a soft subbase of (

(e) For each ^{c}

(f) For each _{pu}_{τ}_{ω} (_{σ}_{pu}

(g) For each

(a) =⇒ (b): Let _{pu} (_{x}) ∊̃_{pu} is soft _{x}. Thus, there exists _{ω} such that _{x}∊̃_{pu}(_{pu}(

(b) =⇒ (c): Obvious.

(c) ⇒ (d): Obvious.

(d) ⇒ (e): Let ^{c}_{pu}_{x}_{B}

So,

By (d),

(e) ⇒ (f): Let _{σ}_{pu}^{c}

and so,

(f) ⇒ (g): Let

Therefore,

(g) ⇒ (a): Let _{x}_{pu}_{x}_{B}^{c}_{x}_{ω}_{x}

If _{pu}_{x}_{a}_{u}_{(}_{a}_{)}) is

Suppose that _{pu}_{x}_{u}_{(}_{a}_{)} such that _{pu}_{x}_{p}_{(}_{x}_{)} ∈ _{pu}_{x}_{ω}_{x}_{pu}_{ω}_{a}_{ω}_{a}_{a}_{ω}_{z}_{pu}_{pu}_{z}_{p}_{(}_{z}_{)} ∊̃_{a}_{u}_{(}_{a}_{)}) is

If _{pu}_{a}_{u}_{(}_{a}_{)}) is

Let _{a}_{u}_{(}_{a}_{)}) is _{pu}_{x}_{a}_{u}_{(}_{a}_{)}) is _{a}_{u}_{(}_{a}_{)}) is

Let _{pu}_{x}

Suppose that _{pu}_{x}_{ω}_{x}_{U}_{ω}_{ω}_{pu}_{U}_{p}_{(}_{U}_{)} ⊂̃_{pu}_{x}

Suppose that _{pu}_{p}_{x}

_{a}_{u}_{(}_{a}_{)}) is _{a}_{u}_{(}_{a}_{)} = ℵ,

Let _{pu}

Soft continuous soft functions are soft

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:

Let _{pu}

If _{pu}_{qv}_{(}_{q}_{◦}_{p}_{)(}_{v}_{◦}_{u}_{)} : (

Let _{qv}_{pu}_{(}_{q}_{◦}_{p}_{)(}_{v}_{◦}_{u}_{)} : (

If _{pu}_{p}_{|Z})_{u}_{Z}

Let _{pu}_{p}_{|Z})_{u}_{Z}

Let _{pu}_{A}_{Z}_{W}_{Z}_{W}^{c}_{A}_{p}_{|Z})_{u}_{Z}_{p}_{|W})_{u}_{W}_{pu}

We apply statement (e) of Theorem 2.2. Let ^{c}

Because _{p}_{|Z})_{u}_{p}_{|W})_{u}

Therefore, by Theorem 17 of [2], _{Z}_{W}^{c}

Let _{pu}_{x}_{Z}_{ω}_{x}_{Z}_{p}_{|Z})_{u}_{Z}_{x}_{pu}_{x}

Let _{pu}_{x}_{p}_{|Z})_{u}_{x}_{Z}_{ω}_{x}_{p}_{|Z})_{u}_{Z}_{ω}_{Z}_{ω}_{ω}_{pu}_{x}

_{pu}_{X}_{α} : _{A}_{X}_{α} : _{p}_{|}_{X}_{α})^{u}_{α}_{X}_{α}, _{pu}

Let _{x}_{pu}_{x}_{A}_{X}_{α} : _{∘} ∈ Δsuch that _{x}_{X}_{α∘}. Therefore, by Theorem 2.12, it follows that _{pu}

In this section, we introduce the concept of a soft _{s}_{s}

A soft function _{pu}_{s}

Soft continuous functions are soft _{s}

Let _{pu}_{pu}_{pu}_{s}

Soft _{s}

Let _{pu}_{s}_{s}_{pu}_{pu}

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

Let _{A}_{A}_{ℕ}, _{ℚc}, _{(ℕ∪ℚc})}, and _{B}_{B}_{{}_{a}_{}}, _{{}_{b}_{}}}. It is not difficult to see that _{τ}_{ω} (_{ℕ}) = _{ℕ}, _{τ}_{ℕ}) = _{ℚ}, and _{τ}_{ω} (_{ℚc}) = _{ℝ−ℕ}. Thus, _{ℚ} ∈ _{s}_{ℝ−ℕ} ∈ _{s}

Because _{pu}_{s}_{qv}_{s}

If _{pu}_{s}_{pu}

Let _{s}_{pu}_{pu}

If _{pu}_{pu}_{s}

Let _{pu}_{pu}_{s}

For a soft function _{pu}

(a) _{pu}_{s}

(b) For each _{x}_{pu}_{x}_{s}_{x}_{pu}

(a) ⇒ (b): Assume that _{pu}_{s}_{pu}_{x}_{pu}_{s}_{s}_{x}_{pu}

(b) ⇒ (a): Let _{pu}_{x}_{a}_{x} ∈ _{s}_{x}_{a}_{x} and _{pu}_{a}_{x} )⊂̃

For a soft function _{pu}

(a) _{pu}_{s}

(b) For a soft base

(c) For each _{c}_{s}

(d) For each

(e) For each

(f) For each

(a) ⇒ (b): It is obvious.

(b) ⇒(c): Let ^{c}_{B}_{B}_{1} ⊆ ℳ such that 1_{B}_{1}}. Then

By (b), _{s}

(c) ⇒ (d): Let _{σ}_{pu}^{c}_{s}_{s}

Thus,

(d) ⇒ (e): Let

Therefore,

(e) ⇒ (f): Let

Thus,

(f) ⇒ (a): Let _{σ}

Let (

Because _{s}_{τ}_{s}_{τ}_{ω} (_{τ}_{s}_{τ}_{s}_{τ}_{τ}_{ω}(_{τ}_{τ}_{ω}(_{τ}_{s}_{τ}_{s}_{τ}_{τ}_{ω} (_{τ}_{s}_{τ}_{s}_{τ}_{τ}_{ω} (_{τ}_{τ}_{ω} (_{τ}_{s}_{τ}_{ω} (_{τ}_{τ}

Thus,

Therefore, by Theorem 2.21 of [3], _{τ}_{ω} (_{τ}_{s}

For a soft function _{pu}

(a) _{pu}_{s}

(b) For each

(c) For each

(a) ⇒ (b): Suppose that _{pu}_{s}

_{pu}_{s}_{τ}_{σ}_{pu}

(b) ⇒(a): We apply Theorem 3.8(d). Let

Thus, by Lemma 3.9,

(a) ⇒ (c): Suppose that _{pu}_{s}

(c) ⇒ (a): We apply Theorem 3.8(e). Let

If _{pu}_{s}_{qv}_{(}_{q}_{∘}_{p}_{)(}_{v}_{∘}_{u}_{)} : (_{s}

Let _{qv}_{pu}_{s}_{(}_{q}_{∘}_{p}_{)(}_{v}_{∘}_{u}_{)} : (_{s}

The composition of two soft _{s}_{s}

Let ℑ be the usual topology on ℝ and let

ℝ → ℝ and

Then

and

_{pu}_{qv}_{u}_{pu}_{qv}_{s}_{(2, ∞)}. Then _{qv}_{pu}^{−1} (_{s}_{qv}_{pu}_{s}

Let (_{s}_{s}

For each _{a}

Let _{s}_{pu}_{s}

Suppose that _{s}^{−1}(_{s}

Suppose that _{pu}_{p}_{V}

The classes of _{s}_{s}

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

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E-mail: algore@just.edu.jo

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

Copyright © The Korean Institute of Intelligent Systems.

Samer Al Ghour

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

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

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.

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

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 _{A}_{A}_{A}_{A}^{c}_{s}_{s}_{s}_{s}_{s}_{s}_{s}

Let (_{τ}_{τ}_{ℑ} (

The following definitions and results are used in the following:

Let

(a) _{Y}

(b) [1] _{Y}

(c) _{x}

The set of all soft points in

Let_{x}_{x}_{x}_{x}_{x}

Let (_{a}

Let (

defines a soft topology on

Let _{a}

Then

An STS (

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

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

Let (

(a) [33] semi-open if there exists _{ℑ} (

(b) [34] _{s}_{ℑ}_{ω}_{s}_{s}

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

A function _{ω}

A function ^{−1}(

A function _{s}^{−1}(_{s}

Let (

(a) [30] _{τ}

(b) [30] _{A}

(c) [3] _{s}_{τω}_{s}_{s}

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].

A soft function _{pu}

Let (

(a) The soft _{s}_{s-}_{τ}

(b) The soft _{s}_{s}_{τ}

In this section, we introduce the concept of soft

A soft function _{pu}_{x}_{pu}_{x}_{ω}_{x}_{pu}_{pu}_{x}_{pu}

For a soft function _{pu}

(a) _{pu}

(b) For each

(c) For a soft base ℳ of (

(d) For a soft subbase of (

(e) For each ^{c}

(f) For each _{pu}_{τ}_{ω} (_{σ}_{pu}

(g) For each

(a) =⇒ (b): Let _{pu} (_{x}) ∊̃_{pu} is soft _{x}. Thus, there exists _{ω} such that _{x}∊̃_{pu}(_{pu}(

(b) =⇒ (c): Obvious.

(c) ⇒ (d): Obvious.

(d) ⇒ (e): Let ^{c}_{pu}_{x}_{B}

So,

By (d),

(e) ⇒ (f): Let _{σ}_{pu}^{c}

and so,

(f) ⇒ (g): Let

Therefore,

(g) ⇒ (a): Let _{x}_{pu}_{x}_{B}^{c}_{x}_{ω}_{x}

If _{pu}_{x}_{a}_{u}_{(}_{a}_{)}) is

Suppose that _{pu}_{x}_{u}_{(}_{a}_{)} such that _{pu}_{x}_{p}_{(}_{x}_{)} ∈ _{pu}_{x}_{ω}_{x}_{pu}_{ω}_{a}_{ω}_{a}_{a}_{ω}_{z}_{pu}_{pu}_{z}_{p}_{(}_{z}_{)} ∊̃_{a}_{u}_{(}_{a}_{)}) is

If _{pu}_{a}_{u}_{(}_{a}_{)}) is

Let _{a}_{u}_{(}_{a}_{)}) is _{pu}_{x}_{a}_{u}_{(}_{a}_{)}) is _{a}_{u}_{(}_{a}_{)}) is

Let _{pu}_{x}

Suppose that _{pu}_{x}_{ω}_{x}_{U}_{ω}_{ω}_{pu}_{U}_{p}_{(}_{U}_{)} ⊂̃_{pu}_{x}

Suppose that _{pu}_{p}_{x}

_{a}_{u}_{(}_{a}_{)}) is _{a}_{u}_{(}_{a}_{)} = ℵ,

Let _{pu}

Soft continuous soft functions are soft

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:

Let _{pu}

If _{pu}_{qv}_{(}_{q}_{◦}_{p}_{)(}_{v}_{◦}_{u}_{)} : (

Let _{qv}_{pu}_{(}_{q}_{◦}_{p}_{)(}_{v}_{◦}_{u}_{)} : (

If _{pu}_{p}_{|Z})_{u}_{Z}

Let _{pu}_{p}_{|Z})_{u}_{Z}

Let _{pu}_{A}_{Z}_{W}_{Z}_{W}^{c}_{A}_{p}_{|Z})_{u}_{Z}_{p}_{|W})_{u}_{W}_{pu}

We apply statement (e) of Theorem 2.2. Let ^{c}

Because _{p}_{|Z})_{u}_{p}_{|W})_{u}

Therefore, by Theorem 17 of [2], _{Z}_{W}^{c}

Let _{pu}_{x}_{Z}_{ω}_{x}_{Z}_{p}_{|Z})_{u}_{Z}_{x}_{pu}_{x}

Let _{pu}_{x}_{p}_{|Z})_{u}_{x}_{Z}_{ω}_{x}_{p}_{|Z})_{u}_{Z}_{ω}_{Z}_{ω}_{ω}_{pu}_{x}

_{pu}_{X}_{α} : _{A}_{X}_{α} : _{p}_{|}_{X}_{α})^{u}_{α}_{X}_{α}, _{pu}

Let _{x}_{pu}_{x}_{A}_{X}_{α} : _{∘} ∈ Δsuch that _{x}_{X}_{α∘}. Therefore, by Theorem 2.12, it follows that _{pu}

In this section, we introduce the concept of a soft _{s}_{s}

A soft function _{pu}_{s}

Soft continuous functions are soft _{s}

Let _{pu}_{pu}_{pu}_{s}

Soft _{s}

Let _{pu}_{s}_{s}_{pu}_{pu}

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

Let _{A}_{A}_{ℕ}, _{ℚc}, _{(ℕ∪ℚc})}, and _{B}_{B}_{{}_{a}_{}}, _{{}_{b}_{}}}. It is not difficult to see that _{τ}_{ω} (_{ℕ}) = _{ℕ}, _{τ}_{ℕ}) = _{ℚ}, and _{τ}_{ω} (_{ℚc}) = _{ℝ−ℕ}. Thus, _{ℚ} ∈ _{s}_{ℝ−ℕ} ∈ _{s}

Because _{pu}_{s}_{qv}_{s}

If _{pu}_{s}_{pu}

Let _{s}_{pu}_{pu}

If _{pu}_{pu}_{s}

Let _{pu}_{pu}_{s}

For a soft function _{pu}

(a) _{pu}_{s}

(b) For each _{x}_{pu}_{x}_{s}_{x}_{pu}

(a) ⇒ (b): Assume that _{pu}_{s}_{pu}_{x}_{pu}_{s}_{s}_{x}_{pu}

(b) ⇒ (a): Let _{pu}_{x}_{a}_{x} ∈ _{s}_{x}_{a}_{x} and _{pu}_{a}_{x} )⊂̃

For a soft function _{pu}

(a) _{pu}_{s}

(b) For a soft base

(c) For each _{c}_{s}

(d) For each

(e) For each

(f) For each

(a) ⇒ (b): It is obvious.

(b) ⇒(c): Let ^{c}_{B}_{B}_{1} ⊆ ℳ such that 1_{B}_{1}}. Then

By (b), _{s}

(c) ⇒ (d): Let _{σ}_{pu}^{c}_{s}_{s}

Thus,

(d) ⇒ (e): Let

Therefore,

(e) ⇒ (f): Let

Thus,

(f) ⇒ (a): Let _{σ}

Let (

Because _{s}_{τ}_{s}_{τ}_{ω} (_{τ}_{s}_{τ}_{s}_{τ}_{τ}_{ω}(_{τ}_{τ}_{ω}(_{τ}_{s}_{τ}_{s}_{τ}_{τ}_{ω} (_{τ}_{s}_{τ}_{s}_{τ}_{τ}_{ω} (_{τ}_{τ}_{ω} (_{τ}_{s}_{τ}_{ω} (_{τ}_{τ}

Thus,

Therefore, by Theorem 2.21 of [3], _{τ}_{ω} (_{τ}_{s}

For a soft function _{pu}

(a) _{pu}_{s}

(b) For each

(c) For each

(a) ⇒ (b): Suppose that _{pu}_{s}

_{pu}_{s}_{τ}_{σ}_{pu}

(b) ⇒(a): We apply Theorem 3.8(d). Let

Thus, by Lemma 3.9,

(a) ⇒ (c): Suppose that _{pu}_{s}

(c) ⇒ (a): We apply Theorem 3.8(e). Let

If _{pu}_{s}_{qv}_{(}_{q}_{∘}_{p}_{)(}_{v}_{∘}_{u}_{)} : (_{s}

Let _{qv}_{pu}_{s}_{(}_{q}_{∘}_{p}_{)(}_{v}_{∘}_{u}_{)} : (_{s}

The composition of two soft _{s}_{s}

Let ℑ be the usual topology on ℝ and let

ℝ → ℝ and

Then

and

_{pu}_{qv}_{u}_{pu}_{qv}_{s}_{(2, ∞)}. Then _{qv}_{pu}^{−1} (_{s}_{qv}_{pu}_{s}

Let (_{s}_{s}

For each _{a}

Let _{s}_{pu}_{s}

Suppose that _{s}^{−1}(_{s}

Suppose that _{pu}_{p}_{V}

The classes of _{s}_{s}

- Al Ghour, S, and Bin-Saadon, A (2019). On some generated soft topological spaces and soft homogeneity. Heliyon.
*5*. article no. e02061 - Al Ghour, S, and Hamed, W (2020). On two classes of soft sets in soft topological spaces. Symmetry.
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