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Asymptotic dimension of fuzzy metric spaces

Pawel Grzegrzolka

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.

Asymptotic dimension of 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.

Paper Structure

This paper contains 6 sections, 14 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definition of asymptotic dimension of fuzzy metric spaces
  4. Properties of asymptotic dimension of fuzzy metric spaces
  5. Coarse category of fuzzy metric spaces
  6. Examples

Key Result

Proposition \oldthetheorem

Let $(X,M,*)$ be a fuzzy metric space such that $a*b \neq 0$ whenever $a\neq 0$ and $b \neq 0$. Then the union of two bounded sets if bounded.

Theorems & Definitions (53)

  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Example \oldthetheorem
  • Example \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Example \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • ...and 43 more