International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(1): 89-99
Published online March 25, 2022
https://doi.org/10.5391/IJFIS.2022.22.1.89
© The Korean Institute of Intelligent Systems
Samer Al Ghour
Department of Mathematics and Statistics, Jordan University of Science and Technology, Irbid, Jordan
Correspondence to :
Samer Al Ghour (Samer Al Ghour)
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
The soft θω-closure operator is defined as a new soft operator that lies strictly between the usual soft closure and the soft θ-closure. Sufficient conditions are provided for equivalence between the soft θω-closure and usual soft closure operators, and between the soft θω-closure and soft θ-closure operators. Via the soft θω-closure operator, the soft θω-open sets are defined as a new class of soft sets that lies strictly between the class of soft open sets and the class of soft θ-open sets. It is proven that the class of soft θω-open sets form a new soft topology. The soft ω-regularity is characterized via both the soft θω-closure operator and soft θω-open sets. The soft product theorem and several soft mapping theorems are introduced. The correspondence between the soft topology of the soft θω-open sets of soft topological space and their generated topological spaces, and vice versa, are studied. In addition to these, soft θω-continuity as a strong form of soft θ-continuity is introduced and investigated.
Keywords: Soft θ-closure, Soft θ-open, Soft ω-regular, Soft product, Soft θ-continuity, Soft generated soft topological space.
This paper follows the concepts and terminology that appear in [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
The following definitions and results are used throughout this work:
Let
(a) [1]
(b) [1]
(c) [46]
The set of all soft points in
Let
Let (
Let (
defines a soft topology on
Let
Then,
Let
for each (
Let (
Recall that a soft set
An STS (
(a) [50] soft locally indiscrete if every soft open set is soft closed;
(b) [2] soft locally countable if for each
(c) [2] soft anti-locally countable if for every
(d) [6] soft
(e) [6] soft
In this section, we introduce the
Let (
(a) A soft point
(b)
(c)
(d) The family of all soft
Let (
(a) (
(b)
The following is the main definition of this work.
Let (
(a) A soft point
(b)
(c)
(d) The family of all soft
Let (
(a)
(b) If
(c) If
(a) To demonstrate that
(b) Suppose that
(c) Suppose that
Let (
(a)
(b) If
(a) From Theorem 2.4 (a),
(b) Suppose that
Let (
(a)
(b) If
The proof follows from Theorem 2.5 and Theorem 7 of [6].
Let (
(a)
(b) If
The proof follows from Theorem 2.5 and Theorem 6 of [6].
Let (
(a)
(b) If
(a) Using Theorem 2.4(a),
(b) Suppose that
For any STS (
Let
Let (
Let
Let (
(a) If
(b) For each
(c) For each
(d) For each
(e) For each
(a) Let
(b) As
Hence,
(c) Let
(d) Let
This is a contradiction.
(e) follows from (d) and Lemma 2.10.
Let (
(a) 0
(b) The finite soft union of soft
(c) The arbitrary soft intersection of soft
(a) The proof follows from Theorems 2.2(a) and 2.4(b).
(b) We demonstrate that the soft union of two soft
Hence,
(c) Let
For any STS (
(1) According to Theorem 2.12(a), 0
(2) Let
Then, using Theorem 2.12(b), 1
(3) Let
is soft
Let (
Let
Suppose that for each
Every soft open, soft
Let (
Every countable soft open set in an STS is soft
Using Theorem 2(d) of [2], countable soft sets in an STS are soft
For any STS (
(a) (
(b)
(c) For every
(a) ⇒ (b): Suppose that (
(b) ⇒ (c): Suppose that
(c) ⇒ (a): Suppose that
The inclusions in Theorem 2.9 are not equalities in general.
Let for all
Then,
and
Let (
Let (
Suppose that
Let
Suppose that
for all
Suppose that
Let (
For each
Let (
Suppose that
Let
Suppose that
Suppose that
Let (
For each
Let (
Let
Thus, we obtain
Let (
If
Let
Define
For every soft closed set
Let
Let
Let
Let
Let
Let
In this section, we introduce and investigate soft
A soft function
Let
Suppose that
Suppose that
A soft function
Let
Suppose that
Suppose that
Every soft
Let
The converse of Theorem 3.5 is not true in general, as clarified by the following example:
Let and
. Then, as demonstrated in Example 2.4 of [
If
Suppose that
Every soft continuous function is soft
The following two examples demonstrate that soft continuity and soft
Let and
. Then, as proven in Example 2.7 of [
Let and
. Then, as shown in Example 2.8 of [
The following result provides a sufficient condition for a soft
If
Suppose that
Hence,
Let
Let
Let
Using Theorem 3.12, it is sufficient to prove that for every
If
Suppose that
We have defined and investigated the
No potential conflict of interest relevant to this article is reported.
E-mail: algore@just.edu.jo
International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(1): 89-99
Published online March 25, 2022 https://doi.org/10.5391/IJFIS.2022.22.1.89
Copyright © The Korean Institute of Intelligent Systems.
Samer Al Ghour
Department of Mathematics and Statistics, Jordan University of Science and Technology, Irbid, Jordan
Correspondence to:Samer Al Ghour (Samer Al Ghour)
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
The soft θω-closure operator is defined as a new soft operator that lies strictly between the usual soft closure and the soft θ-closure. Sufficient conditions are provided for equivalence between the soft θω-closure and usual soft closure operators, and between the soft θω-closure and soft θ-closure operators. Via the soft θω-closure operator, the soft θω-open sets are defined as a new class of soft sets that lies strictly between the class of soft open sets and the class of soft θ-open sets. It is proven that the class of soft θω-open sets form a new soft topology. The soft ω-regularity is characterized via both the soft θω-closure operator and soft θω-open sets. The soft product theorem and several soft mapping theorems are introduced. The correspondence between the soft topology of the soft θω-open sets of soft topological space and their generated topological spaces, and vice versa, are studied. In addition to these, soft θω-continuity as a strong form of soft θ-continuity is introduced and investigated.
Keywords: Soft &theta,-closure, Soft &theta,-open, Soft &omega,-regular, Soft product, Soft &theta,-continuity, Soft generated soft topological space.
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
The following definitions and results are used throughout this work:
Let
(a) [1]
(b) [1]
(c) [46]
The set of all soft points in
Let
Let (
Let (
defines a soft topology on
Let
Then,
Let
for each (
Let (
Recall that a soft set
An STS (
(a) [50] soft locally indiscrete if every soft open set is soft closed;
(b) [2] soft locally countable if for each
(c) [2] soft anti-locally countable if for every
(d) [6] soft
(e) [6] soft
In this section, we introduce the
Let (
(a) A soft point
(b)
(c)
(d) The family of all soft
Let (
(a) (
(b)
The following is the main definition of this work.
Let (
(a) A soft point
(b)
(c)
(d) The family of all soft
Let (
(a)
(b) If
(c) If
(a) To demonstrate that
(b) Suppose that
(c) Suppose that
Let (
(a)
(b) If
(a) From Theorem 2.4 (a),
(b) Suppose that
Let (
(a)
(b) If
The proof follows from Theorem 2.5 and Theorem 7 of [6].
Let (
(a)
(b) If
The proof follows from Theorem 2.5 and Theorem 6 of [6].
Let (
(a)
(b) If
(a) Using Theorem 2.4(a),
(b) Suppose that
For any STS (
Let
Let (
Let
Let (
(a) If
(b) For each
(c) For each
(d) For each
(e) For each
(a) Let
(b) As
Hence,
(c) Let
(d) Let
This is a contradiction.
(e) follows from (d) and Lemma 2.10.
Let (
(a) 0
(b) The finite soft union of soft
(c) The arbitrary soft intersection of soft
(a) The proof follows from Theorems 2.2(a) and 2.4(b).
(b) We demonstrate that the soft union of two soft
Hence,
(c) Let
For any STS (
(1) According to Theorem 2.12(a), 0
(2) Let
Then, using Theorem 2.12(b), 1
(3) Let
is soft
Let (
Let
Suppose that for each
Every soft open, soft
Let (
Every countable soft open set in an STS is soft
Using Theorem 2(d) of [2], countable soft sets in an STS are soft
For any STS (
(a) (
(b)
(c) For every
(a) ⇒ (b): Suppose that (
(b) ⇒ (c): Suppose that
(c) ⇒ (a): Suppose that
The inclusions in Theorem 2.9 are not equalities in general.
Let for all
Then,
and
Let (
Let (
Suppose that
Let
Suppose that
for all
Suppose that
Let (
For each
Let (
Suppose that
Let
Suppose that
Suppose that
Let (
For each
Let (
Let
Thus, we obtain
Let (
If
Let
Define
For every soft closed set
Let
Let
Let
Let
Let
Let
In this section, we introduce and investigate soft
A soft function
Let
Suppose that
Suppose that
A soft function
Let
Suppose that
Suppose that
Every soft
Let
The converse of Theorem 3.5 is not true in general, as clarified by the following example:
Let and
. Then, as demonstrated in Example 2.4 of [
If
Suppose that
Every soft continuous function is soft
The following two examples demonstrate that soft continuity and soft
Let and
. Then, as proven in Example 2.7 of [
Let and
. Then, as shown in Example 2.8 of [
The following result provides a sufficient condition for a soft
If
Suppose that
Hence,
Let
Let
Let
Using Theorem 3.12, it is sufficient to prove that for every
If
Suppose that
We have defined and investigated the