A New Notion of Fuzzy Function Ideal Convergence

Abstract

P.M. Pu and Y.M. Liu extended Moore-Smith’s convergence of nets to fuzzy topology and Y.M. Liu provided analogous results to J. Kelley’s classical characterization theorem of net convergence by introducing the notion of fuzzy convergence classes. In a previous paper, the authors of this study provided modified versions of this characterization by using an alternative notion of convergence of fuzzy nets, introduced by B.M.U. Afsan, named fuzzy net ideal convergence. Our main scope here is to generalize and simplify the preceding results. Specifically, we insert the concept of a fuzzy function ideal convergence class, L, on a non-empty set, X, consisting of triads (f,e,I), where f is a function from a non-empty set, D, to the set FP(X) of fuzzy points in X, which we call fuzzy function, e∈FP(X), and I is a proper ideal on D, and we provide necessary and sufficient conditions to establish the existence of a unique fuzzy topology, δ, on X, such that (f,e,I)∈L iff fI-converges to e, relative to the fuzzy topology δ.