# Lipschitz homotopy convergence of Alexandrov spaces

## TL;DR

This paper introduces a new concept of good coverings for metric spaces and demonstrates that such spaces share the same Lipschitz homotopy type as their nerve complexes, leading to stability results for Alexandrov spaces.

## Contribution

It defines good coverings for metric spaces and proves their relation to Lipschitz homotopy types, applying this to Alexandrov spaces for stability analysis.

## Key findings

- Spaces with good coverings have the same Lipschitz homotopy type as their nerve complexes.
- Established Lipschitz homotopy stability for a class of Alexandrov spaces.
- Provided a new tool for analyzing the topology of metric spaces via coverings.

## Abstract

We introduce the notion of good coverings of metric spaces, and prove that if a metric space admits a good covering, then it has the same locally Lipschitz homotopy type as the nerve complex of the covering. As an application, we obtain a Lipschitz homotopy stability result for a moduli space of compact Alexandrov spaces without collapsing.