New fuzzy topologies via ideals and generalized openness

Abstract

This paper introduces and investigates a new class of generalized open sets, called fuzzy hI -open sets, in fuzzy ideal topological spaces (X, τ˜, I˜). We prove that the collection of all fuzzy hI -open sets forms a fuzzy topology τ˜ hI satisfying τ˜ ⊆ τ˜ hI and show that τ˜ ∗ and τ˜ hI are in general incomparable, demonstrating that the hI -construction captures fundamentally different information from the ∗-topology. We establish precise conditions under which these topologies coincide and introduce a fuzzy hI -T1 separation axiom. Furthermore, we develop a comprehensive hierarchy of generalizations—fuzzy hαI -open, fuzzy hpI -open, fuzzy hsI -open, and fuzzy hβI -open sets—and prove that these classes are pairwise distinct through genuinely fuzzy (non-characteristic) examples. We introduce fuzzy hI -continuous and fuzzy hI -irresolute functions, providing six equivalent characterizations and a closed-set criterion via the ∗-interior operator. The framework is applied to a concrete multi-criteria decision-making problem, where the ideal filters negligible criteria and the hI -interior provides a refined ranking that demonstrably outperforms the original fuzzy topology