International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(3): 311-317
Published online September 25, 2023
https://doi.org/10.5391/IJFIS.2023.23.3.311
© The Korean Institute of Intelligent Systems
Mitali Routaray^{1}, Prakash Kumar Sahu^{2}, and Sunima Naik^{1}
^{1}Department of Mathematics, School of Applied Sciences, KIIT University, Bhubaneswar, India
^{2}Department of Mathematics, Model Degree College, Nayagarh, Odisha, India
Correspondence to :
Mitali Routaray (mitaray8@gmail.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.
This study introduces the concepts of a fuzzy ℑ^{*}-structure locally compact space and fuzzy ℑ^{*}-structure evaluation maps are introduced and a fuzzy ℑ^{*}-structure covering the map between two fuzzy ℑ^{*}-structure spaces is studied. In addition, the concept of the fuzzy ℑ^{*}-structure exponential law and related theories are demonstrated. Finally, the fuzzy ℑ^{*}-structure compact-open topology is introduced and some of its results are explained.
Keywords: Fuzzy ℑ^{*}-structure space, Fuzzy ℑ^{*}-structure Hausdorff space, Fuzzy ℑ^{*}-structure compact spaces, Fuzzy ℑ^{*}-structure compact-open topology
Fuzzy sets have many applications in engineering, economics, and clinical science. Since then, numerous authors have contributed to the development of a new mathematical field known as “fuzzy mathematics.” Fuzzy topological space theory is a branch of mathematics. In mathematics, the topology provides the most natural framework for fuzzy units to excel. Chang [1] introduced and developed the concept of a fuzzy topological space. The main problem in fuzzy mathematics is applying ordinary concepts to fuzzy cases. Madhuri and Amudhambigai [2] introduced the concept of fuzzy ℑ^{*}-structural spaces related to fuzzy topological spaces. The concept of a compact-open topology is crucial in defining the functional spaces in standard topologies. We aim to introduce the concept of a fuzzy ℑ^{*}-structure compact-open topology and contribute to several theories and effects related to this concept. The concept of the exponential law can be used for the KU algebra [3, 4].
In this study, a new concept of a fuzzy ℑ^{*}-structure locally compact space and fuzzy ℑ^{*}-structure evaluation maps are introduced, and a fuzzy ℑ^{*}-structure covering the map between two fuzzy ℑ^{*}-structure spaces is studied. In addition, the concept of the fuzzy ℑ^{*}-structure exponential law and other theories have been proven. Finally, a fuzzy ℑ^{*}-structure compact-open topology is introduced, and some results of the fuzzy ℑ^{*}-structure compact-open topology are proven. In this study, we established the pivotal concepts of fuzzy local compactness and fuzzy product topology.
In terms of notation, we use letters such as
A function from a nonempty set
Let
(
(
(
The ordered pair (
Let (
Let (
Let (
Let
Let (
A fuzzy topological space is called a fuzzy Hausdorff space if and only if (iff), for any two distinct fuzzy points
A mapping
Let (
Let
If
Let (
Let (
A fuzzy ℑ^{*}-structure space (
Let
Let
The mapping
The product of two fuzzy sets
A fuzzy ℑ^{*}-structure compact subspace of a fuzzy ℑ^{*}-structure Hausdorff space is fuzzy ℑ^{*}-structure closed.
Let (
The following criteria are equivalent in a fuzzy ℑ^{*}-structured Hausdorff space:
(
(
We assume that
Assume that for every fuzzy point
The fuzzy ℑ^{*}-structured Hausdorff space
We assume that
Let (
We give this class
Let
For any
can be as a fuzzy sub base to generate a fuzzy ℑ^{*}-structure topology on the class
We now consider the fuzzy ℑ^{*}-structure product topological space
A fuzzy ℑ^{*}-structure evaluation map is a mapping
Let (
Suppose (
Consider the
Now, we consider the induced map of a given. function
Consider two fuzzy ℑ^{*}-structures,
Without loss of generality, we will suppose that
Any fuzzy point in
Let
We assume that
The fuzzy ℑ^{*}-structure identity function on
implies
By contrast, assume that
We have (
Consequently,
for any
We explored some of the features of the exponential law using induced maps.
Let (
thus
Consider any fuzzy ℑ^{*}-structure subbasis neighborhood
We assert that
The following ℑ^{*}-structure evaluation maps are used to show the fuzzy ℑ^{*}-structure continuity of
The fuzzy ℑ^{*}-structure identity maps on (
Now, checking that (
The fuzzy ℑ^{*}-structure map
Let
Consider the following composition.
where
Let
In this study, the concepts of fuzzy ℑ^{*}-structure compactopen topology and fuzzy ℑ^{*}-structure exponential law were introduced, along with some basic theories. This work is foundational for further research on fuzzy ℑ^{*}-structured compact-open topologies. We aim to build the concept of fuzzy higher homotopy groups and a fuzzy universal covering space using this concept. Furthermore, this topic can be expanded to fuzzy category theory in the future.
No potential conflict of interest relevant to this article was reported.
International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(3): 311-317
Published online September 25, 2023 https://doi.org/10.5391/IJFIS.2023.23.3.311
Copyright © The Korean Institute of Intelligent Systems.
Mitali Routaray^{1}, Prakash Kumar Sahu^{2}, and Sunima Naik^{1}
^{1}Department of Mathematics, School of Applied Sciences, KIIT University, Bhubaneswar, India
^{2}Department of Mathematics, Model Degree College, Nayagarh, Odisha, India
Correspondence to:Mitali Routaray (mitaray8@gmail.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.
This study introduces the concepts of a fuzzy ℑ^{*}-structure locally compact space and fuzzy ℑ^{*}-structure evaluation maps are introduced and a fuzzy ℑ^{*}-structure covering the map between two fuzzy ℑ^{*}-structure spaces is studied. In addition, the concept of the fuzzy ℑ^{*}-structure exponential law and related theories are demonstrated. Finally, the fuzzy ℑ^{*}-structure compact-open topology is introduced and some of its results are explained.
Keywords: Fuzzy &image,^{*}-structure space, Fuzzy &image,^{*}-structure Hausdorff space, Fuzzy &image,^{*}-structure compact spaces, Fuzzy &image,^{*}-structure compact-open topology
Fuzzy sets have many applications in engineering, economics, and clinical science. Since then, numerous authors have contributed to the development of a new mathematical field known as “fuzzy mathematics.” Fuzzy topological space theory is a branch of mathematics. In mathematics, the topology provides the most natural framework for fuzzy units to excel. Chang [1] introduced and developed the concept of a fuzzy topological space. The main problem in fuzzy mathematics is applying ordinary concepts to fuzzy cases. Madhuri and Amudhambigai [2] introduced the concept of fuzzy ℑ^{*}-structural spaces related to fuzzy topological spaces. The concept of a compact-open topology is crucial in defining the functional spaces in standard topologies. We aim to introduce the concept of a fuzzy ℑ^{*}-structure compact-open topology and contribute to several theories and effects related to this concept. The concept of the exponential law can be used for the KU algebra [3, 4].
In this study, a new concept of a fuzzy ℑ^{*}-structure locally compact space and fuzzy ℑ^{*}-structure evaluation maps are introduced, and a fuzzy ℑ^{*}-structure covering the map between two fuzzy ℑ^{*}-structure spaces is studied. In addition, the concept of the fuzzy ℑ^{*}-structure exponential law and other theories have been proven. Finally, a fuzzy ℑ^{*}-structure compact-open topology is introduced, and some results of the fuzzy ℑ^{*}-structure compact-open topology are proven. In this study, we established the pivotal concepts of fuzzy local compactness and fuzzy product topology.
In terms of notation, we use letters such as
A function from a nonempty set
Let
(
(
(
The ordered pair (
Let (
Let (
Let (
Let
Let (
A fuzzy topological space is called a fuzzy Hausdorff space if and only if (iff), for any two distinct fuzzy points
A mapping
Let (
Let
If
Let (
Let (
A fuzzy ℑ^{*}-structure space (
Let
Let
The mapping
The product of two fuzzy sets
A fuzzy ℑ^{*}-structure compact subspace of a fuzzy ℑ^{*}-structure Hausdorff space is fuzzy ℑ^{*}-structure closed.
Let (
The following criteria are equivalent in a fuzzy ℑ^{*}-structured Hausdorff space:
(
(
We assume that
Assume that for every fuzzy point
The fuzzy ℑ^{*}-structured Hausdorff space
We assume that
Let (
We give this class
Let
For any
can be as a fuzzy sub base to generate a fuzzy ℑ^{*}-structure topology on the class
We now consider the fuzzy ℑ^{*}-structure product topological space
A fuzzy ℑ^{*}-structure evaluation map is a mapping
Let (
Suppose (
Consider the
Now, we consider the induced map of a given. function
Consider two fuzzy ℑ^{*}-structures,
Without loss of generality, we will suppose that
Any fuzzy point in
Let
We assume that
The fuzzy ℑ^{*}-structure identity function on
implies
By contrast, assume that
We have (
Consequently,
for any
We explored some of the features of the exponential law using induced maps.
Let (
thus
Consider any fuzzy ℑ^{*}-structure subbasis neighborhood
We assert that
The following ℑ^{*}-structure evaluation maps are used to show the fuzzy ℑ^{*}-structure continuity of
The fuzzy ℑ^{*}-structure identity maps on (
Now, checking that (
The fuzzy ℑ^{*}-structure map
Let
Consider the following composition.
where
Let
In this study, the concepts of fuzzy ℑ^{*}-structure compactopen topology and fuzzy ℑ^{*}-structure exponential law were introduced, along with some basic theories. This work is foundational for further research on fuzzy ℑ^{*}-structured compact-open topologies. We aim to build the concept of fuzzy higher homotopy groups and a fuzzy universal covering space using this concept. Furthermore, this topic can be expanded to fuzzy category theory in the future.