Fuzzy independent topological spaces generated by fuzzy γ∗-open sets and their applications

To the best of our knowledge, till date there are only two directions along which fuzzy topology (general topology) can be compared, one is the generalization of fuzzy topology (such as fuzzy supra topology, generalized fuzzy topology etc.) and the other is the stronger form of fuzzy topology which...

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Published in:Afrika mathematica Vol. 28; no. 5-6; pp. 909 - 928
Main Author: Bhattacharya, B.
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.09.2017
Springer Nature B.V
Subjects:
Applications of Mathematics
Fuzzy sets
History of Mathematical Sciences
Mathematics
Mathematics and Statistics
Mathematics Education
Topology
Fuzzy I-open set
Fuzzy I-open cover
54D30
open set
54C05
Fuzzy
54A40
Fuzzy topology
Fuzzy independent topology (or a non-linear topology)
ISSN:1012-9405, 2190-7668
Online Access:Get full text
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Summary:To the best of our knowledge, till date there are only two directions along which fuzzy topology (general topology) can be compared, one is the generalization of fuzzy topology (such as fuzzy supra topology, generalized fuzzy topology etc.) and the other is the stronger form of fuzzy topology which is known as Alexandroff fuzzy topology. It means that a given topology is always linear. This paper aims to propose the third direction that is a new parallel form of fuzzy topology (or a non-linear topology) called fuzzy independent topology which is neither a generalization nor a stronger form of the given fuzzy topology though it is a unique natural offshoot of the given fuzzy topology but very rare in existence and that has been shown by defining fuzzy γ ∗ -open set in the sense of Dimitrije Andrijevic, by proving that fuzzy γ ∗ -open set and fuzzy open set are completely independent of each other though the collection of fuzzy γ ∗ -open sets are themselves a fuzzy topology therein. At the same time we claim that it is beyond the scope of general topology with the existing generalized open sets in literature and consequently we move one more step towards learning the difference between topology and fuzzy topology. To this end, we study the fundamental properties of the new structure. Also we study some of the basic properties and characterizations of fuzzy γ ∗ -open sets and few of their applications. The investigation enables us to present a new covering property of the given fuzzy topological space and the preservation theory is obtained. Finally to illustrate the advantage of the proposed concept, we compare the obtained results with some already existing ones.
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ISSN:1012-9405
2190-7668
DOI:10.1007/s13370-017-0483-y