Convergence

Abstract

This paper is devoted to the general theory of convergence structures as well as to the special case of fuzzy convergence. Convergence structures are defined in general by means of a covariant set functor ϕ In the classical case, ϕ is the filter functor. In this paper, we are especially interested in the case of ϕ being a fuzzy filter functor. The specific types of convergence structures considered here do not depend only on ϕ For instance, they also depend on a fixed natural transformation η:id → φ from the identity set functor to ϕ as well as on a fixed preordering on each set ϕX such that for each mapping f: X → Y, ϕ f: ϕX → ϕY is isotone. Above all, there are two goals for this paper, namely to get more insight into the fuzzy convergence and to clarify problems concerning compactness, the fuzzy case included. The new notions investigated in this paper are related either to the general theory of convergence structures or to the fuzzy case.