International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(3): 303-310
Published online September 25, 2023
https://doi.org/10.5391/IJFIS.2023.23.3.303
© The Korean Institute of Intelligent Systems
Yeon Seok Eom , Sang Min Yun , and Seok Jong Lee
Department of Mathematics, Chungbuk National University, Cheongju, Korea
Correspondence to :
Seok Jong Lee (sjl@cbnu.ac.kr)
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.
We introduce four types of continuity namely, the concepts of generalized fuzzy (r, s)-continuous, semi-generalized fuzzy (r, s)-continuous, generalized fuzzy (r, s)-semicontinuous, and generalized fuzzy (r, s)-irresolute mappings based on the notion of generalized fuzzy (r, s)-closed sets. We analyzed the properties of these mappings and examined the relationships among these continuities.
Keywords: Generalized fuzzy (r, s)-continuous, Generalized fuzzy (r, s)-irresolute.
The concept of fuzzy sets was introduced by Zadeh [1]. Chang [2] defined fuzzy topological spaces and several authors have since studied and generalized this concept. One such generalization was introduced by Sostak [3], who used the concept of degree of openness. This type of generalization was rephrased by Chattopadhyay et al. [4] and Ramadan [5].
As a further generalization of fuzzy sets, Atanassov [6] introduced the concept of intuitionistic fuzzy sets and Coker [7] defined the intuitionistic fuzzy topological spaces using these sets. Building on the ideas of the degrees of openness and non-openness, Coker and Demirci [8] defined intuitionistic fuzzy topological spaces in Sostak’s sense as a generalization of smooth topological spaces and intuitionistic fuzzy topological spaces. Kim and Lee [9] introduced the concept of generalized fuzzy (
Based on Levine’s concept of a generalized closed set in topological spaces [10], the concept of generalized fuzzy continuous mappings in fuzzy topological spaces was introduced by Balasubramanian and Sundaram [11]. Subsequently, El-Shafei and Zakari [12] introduced the concept of semi-generalized continuous mapping of Chang’s fuzzy topological spaces.
In this paper, we introduce four types of continuity, namely, the concepts of generalized fuzzy (
For the nonstandard definitions and notation, we refer to [9, 13–17]. Let
Let
(1) and .
(2) and .
(3) and .
is said to be an
Let
(1)
(2)
Let
(1)
(2)
(3) cl(int(
(4) int(cl(
Let be a mapping from a SoIFTS
(1) a
(2) a
Let
Let ( ) be a SoIFTS. For each (
Let
Let ( ) be a SoIFTS. For each (
Let
Let ( ) be a SoIFTS. For each (
Let be a mapping from a fuzzy topological space
Based on the notion of generalized fuzzy (
Let be a mapping from a SoIFTS
It is clear that every fuzzy (
Let
and
Define and , by
and
Then, and are SoIFTs on
Let be a mapping from a SoIFTS
Let
Conversely, let
Let be a mapping from a SoIFTS
Since
Thus,
The following example shows that the converse of Theorem 3.5 need not be true.
Let
We define by
Then, is a SoIFT on
Levine [10] introduced
Let ( ) be a SoIFTS and (
Let and be mappings,
Let
Let
and
Define and by
and
Then, , and are SoIFTs on
Let be a mapping from a SoIFTS
Let be a mapping from a SoIFTS
Straightforward.
The following example shows that the converse of the above theorem need not be true.
Let
and
Define and , by
and
Then, and are SoIFTs on
Let be a mapping from a SoIFTS
Straightforward.
Let be a semi-generalized fuzzy (
(1)
(2) sgcl(
(1) Since cl(
Therefore,
(2) Let
Thus, sgcl(
Let and be mapping, and (
Straightforward.
As in Definition 3.7, we define
Let ( ) be a SoIFTS and (
Let be a mapping from a SoIFTS
The converse follows from Theorem 4.2.
Let be a mapping from a SoIFTS
Let be a mapping from a SoIFTS
It is obvious.
The following example shows that the converse of the above theorem need not be true.
Let
and
Define and , by
and
Then, and are SoIFTs on
Let be a mapping from a SoIFTS
Straightforward.
The following example shows that the converse of the above theorem need not be true.
Let
Define and , by
and
Then, and are SoIFTs on
We consider only the non-trival
We now consider the same
Let be a mapping from a SoIFTS
It is obvious.
Let be a generalized fuzzy (
(1)
(2) gscl(
(1) Since cl(
(2) Let
Hence gscl(
Let and be mapping, and (
It is obvious.
Let be a mapping from a SoIFTS
Let be a mapping from a SoIFTS
It is obvious.
Let be a mapping from a SoIFTS
Let
The following example shows that the converse of the above theorem need not be true.
Let
and
Define and , by
and
Then, and are SoIFTs on
Let and be mappings and (
Let
We introduce four types of continuity, i.e., the concepts of generalized fuzzy (
No potential conflict of interest relevant to this article is reported.
Email: math1518@naver.com
E-mail: jivesm@naver.com.
E-mail: sjl@cbnu.ac.kr
International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(3): 303-310
Published online September 25, 2023 https://doi.org/10.5391/IJFIS.2023.23.3.303
Copyright © The Korean Institute of Intelligent Systems.
Yeon Seok Eom , Sang Min Yun , and Seok Jong Lee
Department of Mathematics, Chungbuk National University, Cheongju, Korea
Correspondence to:Seok Jong Lee (sjl@cbnu.ac.kr)
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.
We introduce four types of continuity namely, the concepts of generalized fuzzy (r, s)-continuous, semi-generalized fuzzy (r, s)-continuous, generalized fuzzy (r, s)-semicontinuous, and generalized fuzzy (r, s)-irresolute mappings based on the notion of generalized fuzzy (r, s)-closed sets. We analyzed the properties of these mappings and examined the relationships among these continuities.
Keywords: Generalized fuzzy (r, s)-continuous, Generalized fuzzy (r, s)-irresolute.
The concept of fuzzy sets was introduced by Zadeh [1]. Chang [2] defined fuzzy topological spaces and several authors have since studied and generalized this concept. One such generalization was introduced by Sostak [3], who used the concept of degree of openness. This type of generalization was rephrased by Chattopadhyay et al. [4] and Ramadan [5].
As a further generalization of fuzzy sets, Atanassov [6] introduced the concept of intuitionistic fuzzy sets and Coker [7] defined the intuitionistic fuzzy topological spaces using these sets. Building on the ideas of the degrees of openness and non-openness, Coker and Demirci [8] defined intuitionistic fuzzy topological spaces in Sostak’s sense as a generalization of smooth topological spaces and intuitionistic fuzzy topological spaces. Kim and Lee [9] introduced the concept of generalized fuzzy (
Based on Levine’s concept of a generalized closed set in topological spaces [10], the concept of generalized fuzzy continuous mappings in fuzzy topological spaces was introduced by Balasubramanian and Sundaram [11]. Subsequently, El-Shafei and Zakari [12] introduced the concept of semi-generalized continuous mapping of Chang’s fuzzy topological spaces.
In this paper, we introduce four types of continuity, namely, the concepts of generalized fuzzy (
For the nonstandard definitions and notation, we refer to [9, 13–17]. Let
Let
(1) and .
(2) and .
(3) and .
is said to be an
Let
(1)
(2)
Let
(1)
(2)
(3) cl(int(
(4) int(cl(
Let be a mapping from a SoIFTS
(1) a
(2) a
Let
Let ( ) be a SoIFTS. For each (
Let
Let ( ) be a SoIFTS. For each (
Let
Let ( ) be a SoIFTS. For each (
Let be a mapping from a fuzzy topological space
Based on the notion of generalized fuzzy (
Let be a mapping from a SoIFTS
It is clear that every fuzzy (
Let
and
Define and , by
and
Then, and are SoIFTs on
Let be a mapping from a SoIFTS
Let
Conversely, let
Let be a mapping from a SoIFTS
Since
Thus,
The following example shows that the converse of Theorem 3.5 need not be true.
Let
We define by
Then, is a SoIFT on
Levine [10] introduced
Let ( ) be a SoIFTS and (
Let and be mappings,
Let
Let
and
Define and by
and
Then, , and are SoIFTs on
Let be a mapping from a SoIFTS
Let be a mapping from a SoIFTS
Straightforward.
The following example shows that the converse of the above theorem need not be true.
Let
and
Define and , by
and
Then, and are SoIFTs on
Let be a mapping from a SoIFTS
Straightforward.
Let be a semi-generalized fuzzy (
(1)
(2) sgcl(
(1) Since cl(
Therefore,
(2) Let
Thus, sgcl(
Let and be mapping, and (
Straightforward.
As in Definition 3.7, we define
Let ( ) be a SoIFTS and (
Let be a mapping from a SoIFTS
The converse follows from Theorem 4.2.
Let be a mapping from a SoIFTS
Let be a mapping from a SoIFTS
It is obvious.
The following example shows that the converse of the above theorem need not be true.
Let
and
Define and , by
and
Then, and are SoIFTs on
Let be a mapping from a SoIFTS
Straightforward.
The following example shows that the converse of the above theorem need not be true.
Let
Define and , by
and
Then, and are SoIFTs on
We consider only the non-trival
We now consider the same
Let be a mapping from a SoIFTS
It is obvious.
Let be a generalized fuzzy (
(1)
(2) gscl(
(1) Since cl(
(2) Let
Hence gscl(
Let and be mapping, and (
It is obvious.
Let be a mapping from a SoIFTS
Let be a mapping from a SoIFTS
It is obvious.
Let be a mapping from a SoIFTS
Let
The following example shows that the converse of the above theorem need not be true.
Let
and
Define and , by
and
Then, and are SoIFTs on
Let and be mappings and (
Let
We introduce four types of continuity, i.e., the concepts of generalized fuzzy (