Asymptotic dimension of fuzzy metric spaces

TL;DR

This paper introduces and studies the asymptotic dimension of fuzzy metric spaces, establishing its invariance and exploring its implications, with calculations for specific examples.

Contribution

It defines the asymptotic dimension for fuzzy metric spaces and proves its invariance, extending classical results to the fuzzy setting.

Findings

Asymptotic dimension is invariant in the fuzzy metric space category.

Several properties of asymptotic dimension mirror the classical metric case.

Calculated asymptotic dimension for specific fuzzy metric spaces.

Abstract

In this paper, we define asymptotic dimension of fuzzy metric spaces in the sense of George and Veeramini. We prove that asymptotic dimension is an invariant in the coarse category of fuzzy metric spaces. We also show several consequences of asymptotic dimension in the fuzzy setting which resemble the consequences of asymptotic dimension in the metric setting. We finish with calculating asymptotic dimension of a few fuzzy metric spaces.