Properties of several fuzzy set spaces

TL;DR

This paper investigates the properties of fuzzy set spaces with specific metrics, exploring their relations, characterizations, and completions to deepen understanding of their topological structure.

Contribution

It provides new relations among various metrics and characterizations of compactness, boundedness, and completions in fuzzy set spaces with endograph and sendograph metrics.

Findings

Relations among endograph, sendograph, and $ extGamma$-convergence

Characterizations of compactness and boundedness in fuzzy set spaces

Descriptions of the completions of these metric spaces

Abstract

This paper discusses the properties the spaces of fuzzy sets in a metric space equipped with the endograph metric and the sendograph metric, respectively. We fist give some relations among the endograph metric, the sendograph metric and the $\Gamma$-convergence, and then investigate the level characterizations of the endograph metric and the $\Gamma$-convergence. By using the above results, we give some relations among the endograph metric, the sendograph metric, the supremum metric and the $d_p^*$ metric, $p\geq 1$. On the basis of the above results, we present the characterizations of total boundedness, relative compactness and compactness in the space of compact positive $\al$-cuts fuzzy sets equipped with the endograph metric, and in the space of compact support fuzzy sets equipped with the sendograph metric, respectively. Furthermore, we give completions of these metric spaces, respectively.