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
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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