A countable set derived by fuzzy set

TL;DR

This paper demonstrates that for any fuzzy set on real space, a related set is at most countable, and it refines a proof concerning compactness in fuzzy set spaces with an $L_p$ metric.

Contribution

It introduces a new result about the countability of a set derived from fuzzy sets and refines existing proofs in fuzzy set space theory.

Findings

The set D(u) associated with any fuzzy set u on R^m is at most countable.

Provides a modified proof of a key assertion in a theorem about compactness in fuzzy set spaces.

Enhances understanding of the structure of fuzzy sets in metric spaces.

Abstract

In this paper, it shows that for each fuzzy set $u$ on $\mathbb{R}^m$, the set $D(u)$ is at most countable. Based on this, it modifies the proof of assertion (I) in step 2 of the sufficiency part of Theorem 4.1 in paper: Characterizations of compact sets in fuzzy sets spaces with $L_p$ metric, http://arxiv.org/abs/1509.00447.