Some notes on characterizations of compact sets in fuzzy number spaces

TL;DR

This paper characterizes compact sets in fuzzy number spaces with level convergence topology, showing their equivalence to sequential compactness, and clarifies discrepancies with previous incorrect characterizations using counterexamples.

Contribution

It provides a new correct characterization of compact sets in fuzzy number spaces with level convergence topology, resolving conflicts with prior flawed characterizations.

Findings

Compactness is equivalent to sequential compactness in the space.

Previous characterizations by other authors are incorrect, as shown by counterexamples.

The paper clarifies the relationship between different topologies and compactness in fuzzy number spaces.

Abstract

In this paper, we presents a characterization of compact subsets of the fuzzy number space equipped with the level convergence topology. Based on this, it is shown that compactness is equivalent to sequential compactness on the fuzzy number space equipped with the level convergence topology. Diamond and Kloeden gave a characterization of compact sets in fuzzy number spaces equipped with the supremum metric, Fang and Xue also gave a characterization of compact sets in one-dimensional fuzzy number spaces equipped with supremum metric. The latter characterization is just the one-dimensional case of the former characterization. There exists conflict between the characterization given by us and the characterizations given by the above mentioned authors. We point out the characterizations gave by them is incorrect by a counterexample.