International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(3): 296-302
Published online September 25, 2022
https://doi.org/10.5391/IJFIS.2022.22.3.296
© The Korean Institute of Intelligent Systems
AbdulGawad. A. Q. Al-Qubati1,2 and Mohamed El Sayed2
1Department of Mathematics, Hodeidah University, Hodeidah, Yemen
2Department of Mathematics, College of Science and Arts, Najran University, Najran, Saudi Arabia
Correspondence to :
A. Q. Al-Qubati (gawad196999@yahoo.com)
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, a new concept of generalized intuitionistic fuzzy topological space, called intuitionistic fuzzy b~-door space, is introduced and several characterizations of intuitionistic fuzzy b~-door spaces are analyzed. Many examples were introduced to prove the validity of these concepts. Moreover, certain properties and relationships between intuitionistic fuzzy b~-door spaces and other intuitionistic fuzzy topological spaces were investigated.
Keywords: Intuitionistic fuzzy topology, Intuitionistic fuzzy door space, Intuitionistic fuzzy b~-door space
The notion of an intuitionistic fuzzy set was first defined and further developed by Atanassov [1,2], as a generalization of the fuzzy set proposed by Zadeh [3]. Using the notion of intuitionistic fuzzy sets, Coker [4] introduced intuitionistic fuzzy topological spaces as a generalization of the fuzzy topological spaces proposed by Chang [5]. Several concepts of the fuzzy topological space have been recently extended to intuitionistic fuzzy topological spaces [6–9]. Parameswari and Thangavelu [10] introduced the concept of
The aim of this study is to generalize the concept of open and closed sets, called
Finally, we identify and discuss the relationship between the intuitionistic fuzzy
Section 1 introduces the relative topic, and the basic concepts of the intuitionistic fuzzy sets and intuitionistic fuzzy topological spaces are presented in Section 2. The new concepts for intuitionistic fuzzy
Let T be a non-empty fixed set. An intuitionistic fuzzy set (IFS)
Let
(VI) 0
Further details regarding the operations of the IF-sets, IF points, IF functions, and other concepts used in this study can be found in previous references [6–9,15].
An intuitionistic fuzzy topology (IFT) on a non-empty set
(I) 0
(II) If
(III) If
In this case, pair (
In this study, we introduce the concept of IF
An IFS A of an IFTS (
(i) intuitionistic fuzzy
(ii) intuitionistic fuzzy
Let
Then the family
Also,
Therefore,
Let
Then the family
Also,
Thus, (
Let
Then the family
Let
Clearly, (
Then,
Let
Let
An IF-point
An IFTS
Let
Then, family
Let
Then, family
Let
Let
Then, the family
Then,
For IF
Let
Let
In Example 3.5,
For IF
Let
Example 3.8 will be useful in determining whether the IF sets in Example 3.5 should be replaced the by the IF
Let
An IF
An IFTS
Every IF
Let
The converse of the aforementioned theorem does not need to be true because not all IF
If
Let
The converse of the aforementioned theorem does not need to be true in general; as shown in Example 3.5,
Every IF subspace of an IF
Let (
An IFTS
If
Let
Let
Let
An IFTS (
Let
Let
An IF
Let
An IFTS (
Every IFTS (
Let
Every IF door space (
Clearly.
The converse of the aforementioned theorem does not need to be true in general because not all IF bs-OS or IF bs-CS are IFO or IFC, respectively; see Example 2.6.
Let (
(i) (
(ii) (
(i) ⇒ (ii) Suppose that (
(ii) ⇒ (i) Let
An IF
The following theorem introduces an additional condition for the converse of Theorem 3.14 to be true.
Every IF irreducible submaximal space (
Let
Let (
(i)
(ii)
Let us denote
An IFTS topological space (
Clearly, every intuitionistic fuzzy door space is an intuitionistic fuzzy IFI
An IFTS
Let
(i)
(ii) Every IFS
(i) ⇒ (ii) First, IFS
(ii) ⇒ (i) Let T be an IF
Let
An IF
Let
An IF-quasi-compact image of an IF
Let
IF
Because every IF
In this study, we introduced the concept of IF
No potential conflict of interest relevant to this article was reported.
E-mail: gawad196999@yahoo.com
E-mail: mohammsed@yahoo.com
International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(3): 296-302
Published online September 25, 2022 https://doi.org/10.5391/IJFIS.2022.22.3.296
Copyright © The Korean Institute of Intelligent Systems.
AbdulGawad. A. Q. Al-Qubati1,2 and Mohamed El Sayed2
1Department of Mathematics, Hodeidah University, Hodeidah, Yemen
2Department of Mathematics, College of Science and Arts, Najran University, Najran, Saudi Arabia
Correspondence to:A. Q. Al-Qubati (gawad196999@yahoo.com)
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, a new concept of generalized intuitionistic fuzzy topological space, called intuitionistic fuzzy b~-door space, is introduced and several characterizations of intuitionistic fuzzy b~-door spaces are analyzed. Many examples were introduced to prove the validity of these concepts. Moreover, certain properties and relationships between intuitionistic fuzzy b~-door spaces and other intuitionistic fuzzy topological spaces were investigated.
Keywords: Intuitionistic fuzzy topology, Intuitionistic fuzzy door space, Intuitionistic fuzzy b~-door space
The notion of an intuitionistic fuzzy set was first defined and further developed by Atanassov [1,2], as a generalization of the fuzzy set proposed by Zadeh [3]. Using the notion of intuitionistic fuzzy sets, Coker [4] introduced intuitionistic fuzzy topological spaces as a generalization of the fuzzy topological spaces proposed by Chang [5]. Several concepts of the fuzzy topological space have been recently extended to intuitionistic fuzzy topological spaces [6–9]. Parameswari and Thangavelu [10] introduced the concept of
The aim of this study is to generalize the concept of open and closed sets, called
Finally, we identify and discuss the relationship between the intuitionistic fuzzy
Section 1 introduces the relative topic, and the basic concepts of the intuitionistic fuzzy sets and intuitionistic fuzzy topological spaces are presented in Section 2. The new concepts for intuitionistic fuzzy
Let T be a non-empty fixed set. An intuitionistic fuzzy set (IFS)
Let
(VI) 0
Further details regarding the operations of the IF-sets, IF points, IF functions, and other concepts used in this study can be found in previous references [6–9,15].
An intuitionistic fuzzy topology (IFT) on a non-empty set
(I) 0
(II) If
(III) If
In this case, pair (
In this study, we introduce the concept of IF
An IFS A of an IFTS (
(i) intuitionistic fuzzy
(ii) intuitionistic fuzzy
Let
Then the family
Also,
Therefore,
Let
Then the family
Also,
Thus, (
Let
Then the family
Let
Clearly, (
Then,
Let
Let
An IF-point
An IFTS
Let
Then, family
Let
Then, family
Let
Let
Then, the family
Then,
For IF
Let
Let
In Example 3.5,
For IF
Let
Example 3.8 will be useful in determining whether the IF sets in Example 3.5 should be replaced the by the IF
Let
An IF
An IFTS
Every IF
Let
The converse of the aforementioned theorem does not need to be true because not all IF
If
Let
The converse of the aforementioned theorem does not need to be true in general; as shown in Example 3.5,
Every IF subspace of an IF
Let (
An IFTS
If
Let
Let
Let
An IFTS (
Let
Let
An IF
Let
An IFTS (
Every IFTS (
Let
Every IF door space (
Clearly.
The converse of the aforementioned theorem does not need to be true in general because not all IF bs-OS or IF bs-CS are IFO or IFC, respectively; see Example 2.6.
Let (
(i) (
(ii) (
(i) ⇒ (ii) Suppose that (
(ii) ⇒ (i) Let
An IF
The following theorem introduces an additional condition for the converse of Theorem 3.14 to be true.
Every IF irreducible submaximal space (
Let
Let (
(i)
(ii)
Let us denote
An IFTS topological space (
Clearly, every intuitionistic fuzzy door space is an intuitionistic fuzzy IFI
An IFTS
Let
(i)
(ii) Every IFS
(i) ⇒ (ii) First, IFS
(ii) ⇒ (i) Let T be an IF
Let
An IF
Let
An IF-quasi-compact image of an IF
Let
IF
Because every IF
In this study, we introduced the concept of IF
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