On $ r $-fuzzy soft $ \gamma $-open sets and fuzzy soft $ \gamma $-continuous functions with some applications

Fahad Alsharari; Ahmed O. M. Abubaker; Islam M. Taha; Fahad Alsharari; Ahmed O. M. Abubaker; Islam M. Taha

doi:10.3934/math.2025244

AIMS Mathematics

2025, Volume 10, Issue 3: 5285-5306. doi: 10.3934/math.2025244

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Research article

On $ r $-fuzzy soft $ \gamma $-open sets and fuzzy soft $ \gamma $-continuous functions with some applications

1.
Department of Mathematics, College of Science, Jouf University, Sakaka, 72311, Saudi Arabia
2.
Department of Mathematics, University College of Umluj, University of Tabuk, Saudi Arabia
3.
Department of Mathematics, Faculty of Science, Sohag University, Sohag, Egypt

Received: 03 January 2025 Revised: 10 February 2025 Accepted: 20 February 2025 Published: 10 March 2025
MSC : 54A05, 54A40, 54C05, 54C08, 54D15

In this paper, we defined and discussed a new class of fuzzy soft open (FS-open) sets, called $ r $-fuzzy soft $ \gamma $-open ($ r $-FS-$ \gamma $-open) sets in fuzzy soft topological spaces (FSTSs) based on fuzzy topologies in the sense of Šostak. The class of $ r $-FS-$ \gamma $-open sets is contained in the class of $ r $-FS-$ \beta $-open sets, and contains all $ r $-FS-semi-open and $ r $-FS-pre-open sets. However, we introduced the closure and interior operators with respect to the classes of $ r $-FS-$ \gamma $-closed and $ r $-FS-$ \gamma $-open sets, and studied some of their properties. Thereafter, we defined and studied some new FS-functions using $ r $-FS-$ \gamma $-open and $ r $-FS-$ \gamma $-closed sets, called FS-$ \gamma $-continuous (respectively (resp. for short) FS-$ \gamma $-irresolute, FS-$ \gamma $-open, FS-$ \gamma $-irresolute open, FS-$ \gamma $-closed, and FS-$ \gamma $-irresolute closed) functions. The relationships between these classes of functions were discussed with the help of some illustrative examples. We also explored and established the notions of FS-weakly (resp. FS-almost) $ \gamma $-continuous functions, which are weaker forms of FS-$ \gamma $-continuous functions. We showed that FS-$ \gamma $-continuity $ \Longrightarrow $ FS-almost $ \gamma $-continuity $ \Longrightarrow $ FS-weak $ \gamma $-continuity, but the converse may not be true. After that, we presented some new types of FS-separation axioms, called $ r $-FS-$ \gamma $-regular and $ r $-FS-$ \gamma $-normal spaces using $ r $-FS-$ \gamma $-closed sets, and investigated some properties of them. Finally, we introduced a new type of FS-connectedness, called $ r $-FS-$ \gamma $-connected sets using $ r $-FS-$ \gamma $-closed sets.
Citation: Fahad Alsharari, Ahmed O. M. Abubaker, Islam M. Taha. On $ r $-fuzzy soft $ \gamma $-open sets and fuzzy soft $ \gamma $-continuous functions with some applications[J]. AIMS Mathematics, 2025, 10(3): 5285-5306. doi: 10.3934/math.2025244

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Abstract

In this paper, we defined and discussed a new class of fuzzy soft open (FS-open) sets, called $ r $-fuzzy soft $ \gamma $-open ($ r $-FS-$ \gamma $-open) sets in fuzzy soft topological spaces (FSTSs) based on fuzzy topologies in the sense of Šostak. The class of $ r $-FS-$ \gamma $-open sets is contained in the class of $ r $-FS-$ \beta $-open sets, and contains all $ r $-FS-semi-open and $ r $-FS-pre-open sets. However, we introduced the closure and interior operators with respect to the classes of $ r $-FS-$ \gamma $-closed and $ r $-FS-$ \gamma $-open sets, and studied some of their properties. Thereafter, we defined and studied some new FS-functions using $ r $-FS-$ \gamma $-open and $ r $-FS-$ \gamma $-closed sets, called FS-$ \gamma $-continuous (respectively (resp. for short) FS-$ \gamma $-irresolute, FS-$ \gamma $-open, FS-$ \gamma $-irresolute open, FS-$ \gamma $-closed, and FS-$ \gamma $-irresolute closed) functions. The relationships between these classes of functions were discussed with the help of some illustrative examples. We also explored and established the notions of FS-weakly (resp. FS-almost) $ \gamma $-continuous functions, which are weaker forms of FS-$ \gamma $-continuous functions. We showed that FS-$ \gamma $-continuity $ \Longrightarrow $ FS-almost $ \gamma $-continuity $ \Longrightarrow $ FS-weak $ \gamma $-continuity, but the converse may not be true. After that, we presented some new types of FS-separation axioms, called $ r $-FS-$ \gamma $-regular and $ r $-FS-$ \gamma $-normal spaces using $ r $-FS-$ \gamma $-closed sets, and investigated some properties of them. Finally, we introduced a new type of FS-connectedness, called $ r $-FS-$ \gamma $-connected sets using $ r $-FS-$ \gamma $-closed sets.

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