Exploring ∆-Continuous and ∆-Irresolute Mappings in Topological Spaces

A. In this study, we investigate the concepts of -continuous, -irresolute, -open, and -closed mappings.We establish that every continuous mapping is inherently -irresolute and that every irresolute mapping is -continuous.However, the converse implications do not necessarily hold.This distinction sets -continuous mappings apart from traditional continuous mappings, particularly since the composition of two -continuous mappings may not always preserve -continuity.We propose several methods for constructing new -continuous (or -irresolute) mappings from existing ones, including pasting-type lemmas specifically tailored for these mappings.Additionally, we present counterexamples to illustrate and clarify these concepts.