International Journal of Fuzzy Logic and Intelligent Systems 2021; 21(2): 159-168
Published online June 25, 2021
https://doi.org/10.5391/IJFIS.2021.21.2.159
© 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.
Soft ω_{s}-open sets as a class of soft sets that lies strictly between soft open sets and soft semi-open sets is introduced. The natural properties of soft ω_{s}-open sets are described. Using soft ω_{s}-open sets, soft ω_{s}-closure and soft ω_{s}-interior as new soft operators are defined and investigated. Furthermore, the relationships regarding generated soft topological spaces and generated topological spaces are studied.
Keywords: Soft ω-open sets, Soft semi-open sets, Soft ω_{s}-open, Generated soft topology
Throughout this paper, we follow the notions and terminologies as used in [1] and [2], and for simplicity, STS stands for soft topological space. For the purpose of dealing with uncertain objects, Molodtsov [3] introduced soft sets in 1999. Let
Chen [26] introduced the concept of soft semi-open sets, a weaker form of soft open sets in STSs. This paper was followed by many more papers about soft semi-open sets and their modifications. In this paper, we introduce soft
In [27,28], the authors showed that soft sets are a class of special information systems. This insight constitutes our motivation to study the structures of soft sets for information systems. Therefore, this paper does not only form the theoretical basis for further applications of soft topology, such as soft
The following definitions and results will be used in the sequel:
Let (
(a) [29] semi-open if there is
(b) [30]
Let (
(a)
(b)
Let (
(a) The soft semi-closure of
(b) The soft semi-interior of
As defined in [2], a STS (
Let (
A STS (
For any STS (
Let (
(a)
(b)
Let (
Let (
Let
The following three examples show that none of the two inclusions in Theorem 2.2 is equality in general:
Let
Let
Let
Let
If (
By Theorem 2.2, we only need to see that
For any STS (
Let (
If (
By Theorem 2.2, we only need to see that
Soft
Let
Let (
Suppose that
Suppose that
Let (
For every
Soft intersection of two soft
Let
Let (
Let
Let (
Suppose that
Therefore,
For any STS (
As
Let (
Let (
(a) Every soft closed set in (
(b) Every soft
(c) 0
(a) Let
(b) Let
(c) Follows by (a).
One can use Example 2.3 to show that none of the implications in Theorem 2.15 (a), (b) is reversible in general.
Let (
Suppose that
Soft
Consider (
Let (
Suppose that
Let (
Assume that
Suppose that
Therefore, by Theorem 2.10, 1
If
Let
In Theorem 2.22, we cannot drop the condition ‘soft open mapping’:
Let
Let (
Let (
(a)
(b)
(a) Follows from Definition 3.1 and Theorem 2.18.
(b) Follows immediately by (a).
Let (
Follows from the definitions and parts (a) and (b) of Theorem 2.17.
In Theorem 3.3
Consider Example 2.3. We proved that 1
Let (
(a) If
(b) If
(a) Suppose that soft
(b) Suppose that
If (
As (
If (
As (
If (
By Theorem 2.7 we have
Let (
(a) If
(b)
(c)
(a) Suppose that
(b) As
(c) As
The soft inclusion in Theorem 3.9 (b) cannot be replaced by soft equality in general:
Consider (
If (
As (
Let (
By Theorem 3.9 (b) we only need to show that
The soft inclusion in Theorem 3.9 (c) cannot be replaced by soft equality in general:
Let
Let (
As
By Theorem 3.3,
Let (
As
By Theorem 3.3,
Let (
Suppose that
By contradiction. Then we have
Let (
Let (
(a)
(b)
(a) Follows from Definition 3.17 and Theorem 2.11.
(b) Follows immediately by (a).
Let (
Follows from the definitions and Theorem 2.2.
In Theorem 3.19
Consider Example 2.3. We proved that 1
Let (
(a) If
(b) If
(a) Suppose that
(b) Suppose that
If (
As (
If (
As (
If (
By Theorem 2.7 we have
Let (
(a) If
(b)
(c)
(a) Suppose that
(b) As
(c) As
The soft inclusion in Theorem 3.26 (c) cannot be replaced by soft equality in general:
Consider (
while
If (
As (
Let (
By Theorem 3.25 (c) we only need to show that
The soft inclusion in Theorem 3.25 (b) cannot be replaced by soft equality in general:
Consider (
Let (
As
By Theorem 3.19 we have
Hence,
Let (
As
By Theorem 3.19 we have
So by Theorem 3.25 (a) we have
Let (
(a)
(b) 1
(c) 1
(a) Suppose to the contrary that there exists
(b) It is sufficient to show that
Let
(c) Follows from (a) and (b).
At this stage, we believe that the following four questions are natural:
Let (
Let (
Let (
Let (
We will leave Question 4.3 as an open question. However, in the following example we will give negative answers for Questions 4.1, 4.2, and 4.4:
Let
Let
If we add the condition ‘(
Let (
Let
Let {(
Let
If we add the condition ‘
Let {(
Suppose that
Suppose that
and by Lemma 4.7, (
Let {(
Let
If we add the condition ‘
Let {(
Suppose that
Suppose that
We introduced soft
No potential conflict of interest relevant to this article was reported.
No potential conflict of interest relevant to this article was reported.
E-mail: algore@just.edu.jo
International Journal of Fuzzy Logic and Intelligent Systems 2021; 21(2): 159-168
Published online June 25, 2021 https://doi.org/10.5391/IJFIS.2021.21.2.159
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.
Soft ω_{s}-open sets as a class of soft sets that lies strictly between soft open sets and soft semi-open sets is introduced. The natural properties of soft ω_{s}-open sets are described. Using soft ω_{s}-open sets, soft ω_{s}-closure and soft ω_{s}-interior as new soft operators are defined and investigated. Furthermore, the relationships regarding generated soft topological spaces and generated topological spaces are studied.
Keywords: Soft ω-open sets, Soft semi-open sets, Soft ω_{s}-open, Generated soft topology
Throughout this paper, we follow the notions and terminologies as used in [1] and [2], and for simplicity, STS stands for soft topological space. For the purpose of dealing with uncertain objects, Molodtsov [3] introduced soft sets in 1999. Let
Chen [26] introduced the concept of soft semi-open sets, a weaker form of soft open sets in STSs. This paper was followed by many more papers about soft semi-open sets and their modifications. In this paper, we introduce soft
In [27,28], the authors showed that soft sets are a class of special information systems. This insight constitutes our motivation to study the structures of soft sets for information systems. Therefore, this paper does not only form the theoretical basis for further applications of soft topology, such as soft
The following definitions and results will be used in the sequel:
Let (
(a) [29] semi-open if there is
(b) [30]
Let (
(a)
(b)
Let (
(a) The soft semi-closure of
(b) The soft semi-interior of
As defined in [2], a STS (
Let (
A STS (
For any STS (
Let (
(a)
(b)
Let (
Let (
Let
The following three examples show that none of the two inclusions in Theorem 2.2 is equality in general:
Let
Let
Let
Let
If (
By Theorem 2.2, we only need to see that
For any STS (
Let (
If (
By Theorem 2.2, we only need to see that
Soft
Let
Let (
Suppose that
Suppose that
Let (
For every
Soft intersection of two soft
Let
Let (
Let
Let (
Suppose that
Therefore,
For any STS (
As
Let (
Let (
(a) Every soft closed set in (
(b) Every soft
(c) 0
(a) Let
(b) Let
(c) Follows by (a).
One can use Example 2.3 to show that none of the implications in Theorem 2.15 (a), (b) is reversible in general.
Let (
Suppose that
Soft
Consider (
Let (
Suppose that
Let (
Assume that
Suppose that
Therefore, by Theorem 2.10, 1
If
Let
In Theorem 2.22, we cannot drop the condition ‘soft open mapping’:
Let
Let (
Let (
(a)
(b)
(a) Follows from Definition 3.1 and Theorem 2.18.
(b) Follows immediately by (a).
Let (
Follows from the definitions and parts (a) and (b) of Theorem 2.17.
In Theorem 3.3
Consider Example 2.3. We proved that 1
Let (
(a) If
(b) If
(a) Suppose that soft
(b) Suppose that
If (
As (
If (
As (
If (
By Theorem 2.7 we have
Let (
(a) If
(b)
(c)
(a) Suppose that
(b) As
(c) As
The soft inclusion in Theorem 3.9 (b) cannot be replaced by soft equality in general:
Consider (
If (
As (
Let (
By Theorem 3.9 (b) we only need to show that
The soft inclusion in Theorem 3.9 (c) cannot be replaced by soft equality in general:
Let
Let (
As
By Theorem 3.3,
Let (
As
By Theorem 3.3,
Let (
Suppose that
By contradiction. Then we have
Let (
Let (
(a)
(b)
(a) Follows from Definition 3.17 and Theorem 2.11.
(b) Follows immediately by (a).
Let (
Follows from the definitions and Theorem 2.2.
In Theorem 3.19
Consider Example 2.3. We proved that 1
Let (
(a) If
(b) If
(a) Suppose that
(b) Suppose that
If (
As (
If (
As (
If (
By Theorem 2.7 we have
Let (
(a) If
(b)
(c)
(a) Suppose that
(b) As
(c) As
The soft inclusion in Theorem 3.26 (c) cannot be replaced by soft equality in general:
Consider (
while
If (
As (
Let (
By Theorem 3.25 (c) we only need to show that
The soft inclusion in Theorem 3.25 (b) cannot be replaced by soft equality in general:
Consider (
Let (
As
By Theorem 3.19 we have
Hence,
Let (
As
By Theorem 3.19 we have
So by Theorem 3.25 (a) we have
Let (
(a)
(b) 1
(c) 1
(a) Suppose to the contrary that there exists
(b) It is sufficient to show that
Let
(c) Follows from (a) and (b).
At this stage, we believe that the following four questions are natural:
Let (
Let (
Let (
Let (
We will leave Question 4.3 as an open question. However, in the following example we will give negative answers for Questions 4.1, 4.2, and 4.4:
Let
Let
If we add the condition ‘(
Let (
Let
Let {(
Let
If we add the condition ‘
Let {(
Suppose that
Suppose that
and by Lemma 4.7, (
Let {(
Let
If we add the condition ‘
Let {(
Suppose that
Suppose that
We introduced soft
No potential conflict of interest relevant to this article was reported.
Samer Al Ghour
International Journal of Fuzzy Logic and Intelligent Systems 2021; 21(3): 269-279 https://doi.org/10.5391/IJFIS.2021.21.3.269Samer Al Ghour
International Journal of Fuzzy Logic and Intelligent Systems 2021; 21(1): 57-65 https://doi.org/10.5391/IJFIS.2021.21.1.57