New Fuzzy Topologies via Ideals and Generalized Openness

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.