The fuzzy degree of nondensifiability and applications

Abstract

We introduce the notion of fuzzy degree of nondensifiability (FDND for short) as a non-negative real number that measures the Hausdorff distance between a bounded set of a fuzzy metric space and its closest Peano continuum. The FDND allows us to analyze the precompact subsets of a fuzzy metric space. In fact, we present a characterization of the family of precompact and arc-connected subsets in terms of the FDND, namely, those subsets whose FDND is equal to zero, provided that the fuzzy metric space be of stationary type. As an application of our results, we prove a variant of the Sadovskiĭ fixed point theorem based on the FDND, which generalizes that of Schauder in fuzzy Banach spaces. Likewise, we present an example where the FDND method is applicable but the one based on measures of noncompactness is not.