A new perspective on fuzzy mapping theory with invertedly open and closed mappings

Sandeep Kaur; Alkan Özkan; Faizah D. Alanazi; Sandeep Kaur; Alkan Özkan; Faizah D. Alanazi

doi:10.3934/math.2025043

AIMS Mathematics

2025, Volume 10, Issue 1: 921-931. doi: 10.3934/math.2025043

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A new perspective on fuzzy mapping theory with invertedly open and closed mappings

1.
Department of Mathematics and Statistics, Central University of Punjab, Bathinda, Punjab, India
2.
Department of Mathematics, Faculty of Arts and Sciences, I$ \check{g} $dır University, I$ \check{g} $dır, Turkey
3.
Department of Mathematics, College of Science, Northern Border University, Arar, Saudi Arabia

Received: 29 October 2024 Revised: 01 January 2025 Accepted: 08 January 2025 Published: 15 January 2025
MSC : 54A40, 54C05, 94D05

In fuzzy mapping theories, we examine fuzzy closedness and fuzzy continuity of a mapping $ \phi $, characterized respectively by $ \overline{\phi(M)} \subseteq \phi(\overline{M}) $ and $ \phi(\overline{M}) \subseteq \overline{\phi(M)} $, for every fuzzy set $ M $ in $ V $. Here, $ (V, \tau) $ represents a fuzzy topological space (FTs), where $ V = \{v\} $ denotes a set of points. This reveals a fundamental symmetry between the two mappings in connection with the closure operator. On the other hand, the fuzzy openness of a mapping $ \phi $ is characterized by $ \phi(M^\circ) < (\phi(M))^\circ $ for every fuzzy set $ M $ in $ V $. Considering the above statements, it is logical to explore how fuzzy continuity relates to the interior operator. Building on this, we introduce the notion of the invertedly fuzzy open mapping, defined as $ (\phi(M))^\circ < \phi(M^\circ) $ for any fuzzy set $ M $ in $ V $, and discuss its relationship with fuzzy continuity. In our study, we define and analyze invertedly fuzzy open and invertedly fuzzy closed mappings, along with their respective properties. We also delve into how these mappings connect with fuzzy continuous mappings. Furthermore, we examine a characterization of fuzzy homeomorphism for bijective mappings concerning the interior operator.
- fuzzy sets,
- fuzzy topological spaces,
- fuzzy continuous mappings,
- induced mappings,
- invertedly fuzzy open mappings,
- invertedly fuzzy closed mappings
Citation: Sandeep Kaur, Alkan Özkan, Faizah D. Alanazi. A new perspective on fuzzy mapping theory with invertedly open and closed mappings[J]. AIMS Mathematics, 2025, 10(1): 921-931. doi: 10.3934/math.2025043

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Abstract

In fuzzy mapping theories, we examine fuzzy closedness and fuzzy continuity of a mapping $ \phi $, characterized respectively by $ \overline{\phi(M)} \subseteq \phi(\overline{M}) $ and $ \phi(\overline{M}) \subseteq \overline{\phi(M)} $, for every fuzzy set $ M $ in $ V $. Here, $ (V, \tau) $ represents a fuzzy topological space (FTs), where $ V = \{v\} $ denotes a set of points. This reveals a fundamental symmetry between the two mappings in connection with the closure operator. On the other hand, the fuzzy openness of a mapping $ \phi $ is characterized by $ \phi(M^\circ) < (\phi(M))^\circ $ for every fuzzy set $ M $ in $ V $. Considering the above statements, it is logical to explore how fuzzy continuity relates to the interior operator. Building on this, we introduce the notion of the invertedly fuzzy open mapping, defined as $ (\phi(M))^\circ < \phi(M^\circ) $ for any fuzzy set $ M $ in $ V $, and discuss its relationship with fuzzy continuity. In our study, we define and analyze invertedly fuzzy open and invertedly fuzzy closed mappings, along with their respective properties. We also delve into how these mappings connect with fuzzy continuous mappings. Furthermore, we examine a characterization of fuzzy homeomorphism for bijective mappings concerning the interior operator.

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