Characterizations of compact sets in fuzzy set spaces with Lp metric

Abstract

Compactness criteria in fuzzy set spaces endowed with the Lp metric have been studied for several decades. Total boundedness is a key feature of compactness in metric spaces. However, comparing existing compactness criteria in fuzzy set spaces endowed with the Lp metric with the Arzelà–Ascoli theorem, the latter gives compactness criteria by characterizing totally bounded sets while the former does not characterize totally bounded sets. Currently, compactness criteria are only presented for three particular fuzzy set spaces under assumptions of convexity or star-shapedness. General fuzzy sets have become more important in both theory and applications. Therefore, this paper presents characterizations of totally bounded sets, relatively compact sets, and compact sets in general fuzzy set spaces equipped with the Lp metric, but which do not have any assumptions of convexity or star-shapedness. Subsets of these general sets include common fuzzy sets, such as fuzzy numbers, fuzzy star-shaped numbers with respect to the origin, fuzzy star-shaped numbers, and general fuzzy star-shaped numbers. Existing compactness criteria are stated for fuzzy numbers space, the space of fuzzy star-shaped numbers with respect to the origin, and the space of fuzzy star-shaped numbers endowed with the Lp metric, respectively. Constructing completions of fuzzy set spaces with respect to the Lp metric is a problem closely dependent on characterizing totally bounded sets. Based on characterizations of total boundedness and relatively compactness and some discussion of the convexity and star-shapedness of fuzzy sets, we show that the completions of fuzzy set spaces studied here can be obtained using the Lp extension. We also clarify relationships among the ten fuzzy set spaces studied here—the five pairs of original spaces and their corresponding completions. We show that the subspaces have parallel characterizations of totally bounded sets, relatively compact sets, and compact sets. Finally, we discuss properties of the Lp metric on fuzzy set space as an application of our results, and review compactness criteria proposed in previous work.