International Journal of Fuzzy Logic and Intelligent Systems 2021; 21(3): 269-279
Published online September 25, 2021
https://doi.org/10.5391/IJFIS.2021.21.3.269
© 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 paper, we give characterizations for soft minimal soft open sets in terms of the soft closure operator, and we conclude that soft subsets of soft minimal soft open sets are soft preopen sets. In addition to these, we define soft minimal soft sets and soft minimal soft preopen sets as two new classes of soft sets in soft topological spaces, and we define soft prehomogeneity as a new soft topological property. We give several relationships regarding these new notions and related known soft topological notions. We show that soft minimal soft preopen sets are soft points, and we prove that soft minimal soft sets with non-null soft interiors are soft minimal soft open sets. Moreover, we show that soft prehomogeneous soft topological space that has a soft minimal soft set is soft locally indiscrete. Also, we give several characterizations of soft locally indiscrete soft topological space in terms of soft minimal soft open sets, soft minimal soft sets, soft preopen sets, and soft prehomogeneity. We deal with correspondence between our new soft topological notions and their analogs topological ones. Finally, we raise six open questions.
Keywords: Soft minimal soft open sets, Soft preopen soft sets, Soft locally indiscrete, Prehomogeneity, Generated soft topology
This paper follows the notions and terminologies as appeared in [1] and [2]. In this paper, TS and STS will denote topological space and soft topological space, respectively. Molodtsov [3] defined soft sets in 1999. The soft set theory offers a general mathematical tool for dealing with uncertain objects. Let
In this paper, we give characterizations for soft minimal soft open sets in terms of the soft closure operator, and we conclude that soft subsets of soft minimal soft open sets are soft preopen sets. In addition to these, we define soft minimal soft open sets and soft minimal soft preopen sets as two new classes of soft sets in STSs, and we define soft prehomogeneity. as a new soft topological property. We give several relationships regarding these new notions and related known soft topological notions. We show that soft minimal soft preopen sets are soft points and we prove that soft minimal soft sets with non-null soft interiors are soft minimal soft open sets. Moreover, we show that soft prehomogeneous STS that has a soft minimal soft set is soft locally indiscrete. Also, we give several characterizations of soft locally indiscrete in terms of soft minimal soft open sets, soft minimal soft sets, soft preopen sets, and soft prehomogeneity. We deal with correspondence between our new soft topological notions and their analogs topological ones. Finally, we raise six open questions.
Herein, we recall several related definitions and results.
Let (
Let (
(a) [34] a preopen set in (
(b) [27] a minimal open set in (
(c) [32] a minimal set in (
(d) [32] a minimal preopen set in (
The family of all preopen sets (resp. minimal open sets, minimal sets, minimal preopen sets) in (
A function is said to be
(a) preirresolute if
(b) prehomeomorphism if
A TS (
Let
(a)
(b)
(c) Soft union of
(d) Soft intersection of
(e) The difference of
Let Δ be an arbitrary indexed set and {
(a) The soft union of these soft sets is the soft set denoted by
(b) The soft intersection of these soft sets is the soft set denoted by
Let
will be denoted by
Let
will be denoted by
Let
Let (
Let (
defines a soft topology on
Let (
Let (
(a) [39] a soft preopen set in (
(b) [33] a soft minimal soft open set in (
The family of all soft preopen sets (resp. soft minimal soft open sets) in (
An STS (
A soft mapping
(a) soft preirresolute if
(b) soft prehomeomorphism if
We start this section by the following two characterizations of soft minimal soft open sets:
Let (
(a)
(b) For every
(c) For every
(a) =⇒ (b) Let
(b) =⇒ (c) Let
(c) =⇒(a) Suppose to the contrary that there exists
Let (
Let
For an STS (
(a) (
(b)
(c)
(d)
(a) =⇒ (b) We need only to show that
(b) =⇒ (c) Since
(c)=⇒(d) Let
(d) =⇒ (a) It is sufficient to show that
Let (
Let (
Let (
Suppose that
For any STS (
The following example will show that the inclusion in Corollary 3.6 is not equality, in general:
Let
For every
The following question is natural:
For an STS (
The following example is a negative answer for Question 3.8:
Let
If
The following lemma will be used in the following main result:
Let (
(a) minss(
(b) If
(c) If
(a) Let
(b) Suppose to the contrary that
(c) Suppose that
For an STS (
(a) (
(b) minss(
(c) min (
(d) (
(a) =⇒ (b) By Corollary 3.6 we need only to show that minss(
(b) =⇒(c) By (b), ∪̃{
(c) =⇒(d) By (c) and Theorem 4.11 of [1], min (
(d) =⇒ (a) Let ℬ be a soft base which forms a soft partition of 1
Let (
The following lemma will be necessary for proving the next main result:
Let (
Let
The following theorem shows that soft minimal soft preopen sets are the soft points soft preopen sets:
For any STS (
By the definition of
Since
If
Let
The following example shows that the implication in Theorem 3.15 is not reversible, in general:
Let
Then,
Let (
Suppose that
Let
Now by the above claim we have
Hence,
Suppose that
Let (
Let (
Suppose that
Let (
Let (
(a)
(b)
(a) ⇒ (b) Suppose that
(b) ⇒ (a) Follows directly from Corollary 3.20.
The following is the main concept of this section:
An STS (
For an STS (
(a) (
(b) For any
(c) For any two pairs (
Straightforward.
Soft homogeneous STSs are soft prehomogeneous.
Follows because soft homeomorphisms are soft prehomeomorphism.
Every soft locally indiscrete STSs is soft prehomogeneous.
Let (
and
By Theorem 3.3,
The following example shows that the converse of Remark 4.3 is not true, in general:
Let
The following is an example of a soft prehomogeneous STS that is neither soft homogeneous nor soft locally indiscrete:
Let
Then,
(1) (
(2) (
(3) (
and
It is not difficult to check that
Let (
Suppose that (
Follows from Theorem 4.4.
Let (
Follows from Theorems 3.15 and 4.7.
Let (
Since
In the following result,
If (
Suppose that (
By the end of this section, we raise the following question regarding the converse of Theorem 4.10:
Let (
Let (
Let
Let (
Let
Conversely, let
Let
By Theorem 3.5 of [1], {
Let
Let (
By Proposition 7 of [4],
Let (
Suppose that
The following lemma will be used in the following result:
Let
Let
Let
Suppose that
Suppose that
Let (
For every
Let
Let
Conversely, let
and so (
Let
For every
If
Let
Let
Suppose that
Let {(
The following result gives a partial answer for Question 5.14:
Let (
Suppose that (
Suppose that (
The following three questions are natural:
Let
Let (
Let (
We will leave Questions 5.16 and 5.17 as open questions. On the other hand, the following example gives a negative answer for Question 5.18:
International Journal of Fuzzy Logic and Intelligent Systems 2021; 21(3): 269-279
Published online September 25, 2021 https://doi.org/10.5391/IJFIS.2021.21.3.269
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 paper, we give characterizations for soft minimal soft open sets in terms of the soft closure operator, and we conclude that soft subsets of soft minimal soft open sets are soft preopen sets. In addition to these, we define soft minimal soft sets and soft minimal soft preopen sets as two new classes of soft sets in soft topological spaces, and we define soft prehomogeneity as a new soft topological property. We give several relationships regarding these new notions and related known soft topological notions. We show that soft minimal soft preopen sets are soft points, and we prove that soft minimal soft sets with non-null soft interiors are soft minimal soft open sets. Moreover, we show that soft prehomogeneous soft topological space that has a soft minimal soft set is soft locally indiscrete. Also, we give several characterizations of soft locally indiscrete soft topological space in terms of soft minimal soft open sets, soft minimal soft sets, soft preopen sets, and soft prehomogeneity. We deal with correspondence between our new soft topological notions and their analogs topological ones. Finally, we raise six open questions.
Keywords: Soft minimal soft open sets, Soft preopen soft sets, Soft locally indiscrete, Prehomogeneity, Generated soft topology
This paper follows the notions and terminologies as appeared in [1] and [2]. In this paper, TS and STS will denote topological space and soft topological space, respectively. Molodtsov [3] defined soft sets in 1999. The soft set theory offers a general mathematical tool for dealing with uncertain objects. Let
In this paper, we give characterizations for soft minimal soft open sets in terms of the soft closure operator, and we conclude that soft subsets of soft minimal soft open sets are soft preopen sets. In addition to these, we define soft minimal soft open sets and soft minimal soft preopen sets as two new classes of soft sets in STSs, and we define soft prehomogeneity. as a new soft topological property. We give several relationships regarding these new notions and related known soft topological notions. We show that soft minimal soft preopen sets are soft points and we prove that soft minimal soft sets with non-null soft interiors are soft minimal soft open sets. Moreover, we show that soft prehomogeneous STS that has a soft minimal soft set is soft locally indiscrete. Also, we give several characterizations of soft locally indiscrete in terms of soft minimal soft open sets, soft minimal soft sets, soft preopen sets, and soft prehomogeneity. We deal with correspondence between our new soft topological notions and their analogs topological ones. Finally, we raise six open questions.
Herein, we recall several related definitions and results.
Let (
Let (
(a) [34] a preopen set in (
(b) [27] a minimal open set in (
(c) [32] a minimal set in (
(d) [32] a minimal preopen set in (
The family of all preopen sets (resp. minimal open sets, minimal sets, minimal preopen sets) in (
A function is said to be
(a) preirresolute if
(b) prehomeomorphism if
A TS (
Let
(a)
(b)
(c) Soft union of
(d) Soft intersection of
(e) The difference of
Let Δ be an arbitrary indexed set and {
(a) The soft union of these soft sets is the soft set denoted by
(b) The soft intersection of these soft sets is the soft set denoted by
Let
will be denoted by
Let
will be denoted by
Let
Let (
Let (
defines a soft topology on
Let (
Let (
(a) [39] a soft preopen set in (
(b) [33] a soft minimal soft open set in (
The family of all soft preopen sets (resp. soft minimal soft open sets) in (
An STS (
A soft mapping
(a) soft preirresolute if
(b) soft prehomeomorphism if
We start this section by the following two characterizations of soft minimal soft open sets:
Let (
(a)
(b) For every
(c) For every
(a) =⇒ (b) Let
(b) =⇒ (c) Let
(c) =⇒(a) Suppose to the contrary that there exists
Let (
Let
For an STS (
(a) (
(b)
(c)
(d)
(a) =⇒ (b) We need only to show that
(b) =⇒ (c) Since
(c)=⇒(d) Let
(d) =⇒ (a) It is sufficient to show that
Let (
Let (
Let (
Suppose that
For any STS (
The following example will show that the inclusion in Corollary 3.6 is not equality, in general:
Let
For every
The following question is natural:
For an STS (
The following example is a negative answer for Question 3.8:
Let
If
The following lemma will be used in the following main result:
Let (
(a) minss(
(b) If
(c) If
(a) Let
(b) Suppose to the contrary that
(c) Suppose that
For an STS (
(a) (
(b) minss(
(c) min (
(d) (
(a) =⇒ (b) By Corollary 3.6 we need only to show that minss(
(b) =⇒(c) By (b), ∪̃{
(c) =⇒(d) By (c) and Theorem 4.11 of [1], min (
(d) =⇒ (a) Let ℬ be a soft base which forms a soft partition of 1
Let (
The following lemma will be necessary for proving the next main result:
Let (
Let
The following theorem shows that soft minimal soft preopen sets are the soft points soft preopen sets:
For any STS (
By the definition of
Since
If
Let
The following example shows that the implication in Theorem 3.15 is not reversible, in general:
Let
Then,
Let (
Suppose that
Let
Now by the above claim we have
Hence,
Suppose that
Let (
Let (
Suppose that
Let (
Let (
(a)
(b)
(a) ⇒ (b) Suppose that
(b) ⇒ (a) Follows directly from Corollary 3.20.
The following is the main concept of this section:
An STS (
For an STS (
(a) (
(b) For any
(c) For any two pairs (
Straightforward.
Soft homogeneous STSs are soft prehomogeneous.
Follows because soft homeomorphisms are soft prehomeomorphism.
Every soft locally indiscrete STSs is soft prehomogeneous.
Let (
and
By Theorem 3.3,
The following example shows that the converse of Remark 4.3 is not true, in general:
Let
The following is an example of a soft prehomogeneous STS that is neither soft homogeneous nor soft locally indiscrete:
Let
Then,
(1) (
(2) (
(3) (
and
It is not difficult to check that
Let (
Suppose that (
Follows from Theorem 4.4.
Let (
Follows from Theorems 3.15 and 4.7.
Let (
Since
In the following result,
If (
Suppose that (
By the end of this section, we raise the following question regarding the converse of Theorem 4.10:
Let (
Let (
Let
Let (
Let
Conversely, let
Let
By Theorem 3.5 of [1], {
Let
Let (
By Proposition 7 of [4],
Let (
Suppose that
The following lemma will be used in the following result:
Let
Let
Let
Suppose that
Suppose that
Let (
For every
Let
Let
Conversely, let
and so (
Let
For every
If
Let
Let
Suppose that
Let {(
The following result gives a partial answer for Question 5.14:
Let (
Suppose that (
Suppose that (
The following three questions are natural:
Let
Let (
Let (
We will leave Questions 5.16 and 5.17 as open questions. On the other hand, the following example gives a negative answer for Question 5.18:
Samer Al Ghour
International Journal of Fuzzy Logic and Intelligent Systems 2021; 21(2): 159-168 https://doi.org/10.5391/IJFIS.2021.21.2.159Samer 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