International Journal of Fuzzy Logic and Intelligent Systems 2021; 21(1): 57-65
Published online March 25, 2021
https://doi.org/10.5391/IJFIS.2021.21.1.57
© 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, we introduce a new concept in soft topological spaces, namely, soft ω*-paracompactness, and we provide characterizations thereof. Its connection with other related concepts is also studied. In particular, we show that soft ω*-paracompactness and soft paracompactness are independent of each other. In addition, we study the soft ω*-paracompactness of the soft topological space generated by an indexed family of ω*-paracompact topological spaces.
Keywords: ω-open sets, ω*-paracompact, Soft paracompact, Generated soft topology
Throughout this paper, we use the notions and terminology in [1] and [2]; moreover, TS and STS stand for “topological space” and “soft topological space,” respectively. Recently, classical methods have been applied to several problems in various fields, such as engineering, social sciences, and medical sciences. Soft sets were defined by Molodtsov [3], and they have found numerous applications. Let
The complement of an
In this study, using soft
Herein, we recall several related definitions and results.
Let (
(a) The set of all covers of (
(b) The set of all open covers of (
We recall that if , where (
is called a refinement of ℬ (denoted as
) if for each
, there is
; then,
is called locally finite in (
:
ATS () such that
is locally finite in (
is a refinement of ℬ.
Let (
(a) The set of all soft covers of (
(b) The set of all soft open covers of (
(c) The set of all soft closed covers of (
Let (
(a) ℋ is called soft locally finite in (
(b) ℳ is called a soft refinement of ℋ (denoted as ℳ ⪯̃ ℋ) if for each
An STS (
An STS (
An STS (
Let (
Let (
Let
This study is primarily concerned with the following concept.
An STS (
The following three Lemmas are used in Theorem 3.6 below.
If ℋ is soft locally finite in (
Let
{
We assume that
If ℋ is soft locally finite in (
Let
The following example shows that the converse of the implication in Lemma 3.3 is not true in general.
Let
If ℋ is soft locally finite in (
We assume that ℋ is soft locally finite in (
The following provides two characterizations of soft
Let (
(a) (
(b) For every ℋ ∈
(c) For every ℋ ∈
(a) ⇒ (b): Obvious.
(b) ⇒ (c): We assume that (; then,
. As (
. Letℳ1 = {
ℳ1 ⪯̃ ℋ.
Let , there is
(c) ⇒ (a): Let ℋ ∈ . Then,
, and by (c), there is
such that
is soft locally finite in (
. As ℒ ⪯̃ ℋ, for every
and
(1) {
(2) For every
(3) For every ,
(4) {
(5) {
(6) {
(7) {
1. Follows from Lemma 3.5.
2. We assume toward a contradiction that there are . Then,
. As
, there is
such that
3. such that
and so
Let such that
4. Let is soft locally finite, there is
:
:
}, there is
such that
, for every
5. Let
6. Let
7. This follows directly from (4).
The following characterization of soft paracompact STSs is used in the proof of Theorem 3.8.
Let (
Proof. Straightforward.
Let
We assume that . We set
. Then,
(1) ℳ
(2) ℳ
(3) .
1. As
2. Let
3. Let such that
.
We assume that (.
(1)
(2) is soft locally finite in
(3) .
1. For all
2. Let
We have is soft locally finite in
3. Let with
.
Let (
Proof. Straightforward.
Let (
Let (
Proof. Straightforward.
Let (
We assume that (. By Proposition 2.9, we have (
.
Then, {} ∈
. Let
.
(1) .
(2) is locally finite in (
(3) .
1. As ℋ ⊆ and by Proposition 2.9, it follows that
. In addition, as ℋ ∈
.
2. Let is locally finite in (
3. Let , where
, there is
such that
.
We assume that ( such that
is locally finite in (
. Let
.
(1) ℳ ∈
(2) ℳ is soft locally finite in (
(3) ℳ ⪯̃ ℋ.
(1) As , we have ℳ ⊆
, we have
, and thus
. It follows that ℳ ∈
(2) Let is locally finite in (
:
and
(3) Let . As
, there is
Let
We assume that . We set
. Then,
(1) ℳ
(2) ℳ
(3) .
1. As
2. Let
3. Let such that
.
We assume that (.
(1)
(2) is soft locally finite in
(3) .
1. For all
2. Let is soft locally finite in
3. Let with
.
Herein, we show by examples that soft
Every soft regular soft
Let (
If (
We assume that (
An STS (
Let (
We assume that (
Every strongly soft locally finite STS is soft
We assume that (
Every strongly soft locally countable soft
Let (
If (
We assume toward a contradiction that (
Every soft anti-locally countable soft
We assume that (
Let
Let (
We defined and studied a new concept STSs, soft
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(1): 57-65
Published online March 25, 2021 https://doi.org/10.5391/IJFIS.2021.21.1.57
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, we introduce a new concept in soft topological spaces, namely, soft ω*-paracompactness, and we provide characterizations thereof. Its connection with other related concepts is also studied. In particular, we show that soft ω*-paracompactness and soft paracompactness are independent of each other. In addition, we study the soft ω*-paracompactness of the soft topological space generated by an indexed family of ω*-paracompact topological spaces.
Keywords: ω-open sets, ω*-paracompact, Soft paracompact, Generated soft topology
Throughout this paper, we use the notions and terminology in [1] and [2]; moreover, TS and STS stand for “topological space” and “soft topological space,” respectively. Recently, classical methods have been applied to several problems in various fields, such as engineering, social sciences, and medical sciences. Soft sets were defined by Molodtsov [3], and they have found numerous applications. Let
The complement of an
In this study, using soft
Herein, we recall several related definitions and results.
Let (
(a) The set of all covers of (
(b) The set of all open covers of (
We recall that if , where (
is called a refinement of ℬ (denoted as
) if for each
, there is
; then,
is called locally finite in (
:
ATS () such that
is locally finite in (
is a refinement of ℬ.
Let (
(a) The set of all soft covers of (
(b) The set of all soft open covers of (
(c) The set of all soft closed covers of (
Let (
(a) ℋ is called soft locally finite in (
(b) ℳ is called a soft refinement of ℋ (denoted as ℳ ⪯̃ ℋ) if for each
An STS (
An STS (
An STS (
Let (
Let (
Let
This study is primarily concerned with the following concept.
An STS (
The following three Lemmas are used in Theorem 3.6 below.
If ℋ is soft locally finite in (
Let
{
We assume that
If ℋ is soft locally finite in (
Let
The following example shows that the converse of the implication in Lemma 3.3 is not true in general.
Let
If ℋ is soft locally finite in (
We assume that ℋ is soft locally finite in (
The following provides two characterizations of soft
Let (
(a) (
(b) For every ℋ ∈
(c) For every ℋ ∈
(a) ⇒ (b): Obvious.
(b) ⇒ (c): We assume that (; then,
. As (
. Letℳ1 = {
ℳ1 ⪯̃ ℋ.
Let , there is
(c) ⇒ (a): Let ℋ ∈ . Then,
, and by (c), there is
such that
is soft locally finite in (
. As ℒ ⪯̃ ℋ, for every
and
(1) {
(2) For every
(3) For every ,
(4) {
(5) {
(6) {
(7) {
1. Follows from Lemma 3.5.
2. We assume toward a contradiction that there are . Then,
. As
, there is
such that
3. such that
and so
Let such that
4. Let is soft locally finite, there is
:
:
}, there is
such that
, for every
5. Let
6. Let
7. This follows directly from (4).
The following characterization of soft paracompact STSs is used in the proof of Theorem 3.8.
Let (
Proof. Straightforward.
Let
We assume that . We set
. Then,
(1) ℳ
(2) ℳ
(3) .
1. As
2. Let
3. Let such that
.
We assume that (.
(1)
(2) is soft locally finite in
(3) .
1. For all
2. Let
We have is soft locally finite in
3. Let with
.
Let (
Proof. Straightforward.
Let (
Let (
Proof. Straightforward.
Let (
We assume that (. By Proposition 2.9, we have (
.
Then, {} ∈
. Let
.
(1) .
(2) is locally finite in (
(3) .
1. As ℋ ⊆ and by Proposition 2.9, it follows that
. In addition, as ℋ ∈
.
2. Let is locally finite in (
3. Let , where
, there is
such that
.
We assume that ( such that
is locally finite in (
. Let
.
(1) ℳ ∈
(2) ℳ is soft locally finite in (
(3) ℳ ⪯̃ ℋ.
(1) As , we have ℳ ⊆
, we have
, and thus
. It follows that ℳ ∈
(2) Let is locally finite in (
:
and
(3) Let . As
, there is
Let
We assume that . We set
. Then,
(1) ℳ
(2) ℳ
(3) .
1. As
2. Let
3. Let such that
.
We assume that (.
(1)
(2) is soft locally finite in
(3) .
1. For all
2. Let is soft locally finite in
3. Let with
.
Herein, we show by examples that soft
Every soft regular soft
Let (
If (
We assume that (
An STS (
Let (
We assume that (
Every strongly soft locally finite STS is soft
We assume that (
Every strongly soft locally countable soft
Let (
If (
We assume toward a contradiction that (
Every soft anti-locally countable soft
We assume that (
Let
Let (
We defined and studied a new concept STSs, soft
Samer Al Ghour
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