A study on convergence and ideal convergence classes

A New Notion of Fuzzy Function Ideal Convergence

“…Specifically, we examine the necessary and sufficient conditions that a fuzzy function ideal convergence class L, on a non-empty set X, should fulfill to determine a unique fuzzy topology δ on X such that I-convergence (L) coincides with I-convergence, with respect to δ. All the results obtained here are parallel to and extend those given in [18,19,21], for the ordinary topology, while simultaneously simplify the exposition and the underlying theory. In order to increase the utility of the present work, future research options may include the extension of the lattice background, L, to completely distributive lattices with an ordering-reversing involution as a tool to investigate the notion of L-fuzzy function ideal convergence and its applications in the more general context of L-fuzzy topological spaces.…”

“…More precisely, in [18] they considered a non-empty set, X, and a class, C, consisting of triples of the form ((s d ) d∈D , x, I), where (s d ) d∈D is a net with a domain on the directed set, D, and values on X, I is a D-admissible ideal on D and x ∈ X and provided a set of axioms on the class, C, which are necessary and sufficient to ensure the existence of a unique topology, τ, on X, such that ((s d ) d∈D , x, I) ∈ C iff (s d ) d∈D I-converges to x, relative to the topology, τ. Subsequently, in [19], they provided similar results by considering arbitrary ideals. In continuation, in [21] they extended and simplified the results of the last two papers by considering functions instead of nets and a smaller set of axioms, to be fulfilled by class C, in order for the last to be topological.…”

“…Finally, the authors of [19] obtained the following results for the ideal convergence of fuzzy nets. Definition 6.…”

“…In Section 3, we present some special properties concerning the convergence of a fuzzy function, with respect to an ideal, on fuzzy topological spaces. Finally, in Section 4, we consider the notion of a fuzzy function ideal convergence class on a non-empty set, X, and prove analogous to the results of [19] for the more general case of fuzzy function ideal convergence.…”

“…Lahiri and P. Das, in [16,17], investigated the notion of the ideal convergence of sequences and nets in topological spaces. In the context of net ideal convergence, the authors of [18,19] provided a modified version of J. Kelley's classical theorem [20] (p. 74, Theorem 9) for convergence classes. More precisely, in [18] they considered a non-empty set, X, and a class, C, consisting of triples of the form ((s d ) d∈D , x, I), where (s d ) d∈D is a net with a domain on the directed set, D, and values on X, I is a D-admissible ideal on D and x ∈ X and provided a set of axioms on the class, C, which are necessary and sufficient to ensure the existence of a unique topology, τ, on X, such that ((s d ) d∈D , x, I) ∈ C iff (s d ) d∈D I-converges to x, relative to the topology, τ.…”