# Topological spaces of persistence modules and their properties

## TL;DR

This paper systematically studies the topological spaces formed by classes of persistence modules in topological data analysis, focusing on their properties under interleaving distance.

## Contribution

It provides a comprehensive analysis of the topological structures of various classes of persistence modules and their interrelations.

## Key findings

- Interleaving distance induces a topology on classes of persistence modules.
- The paper characterizes basic topological properties of these spaces.
- Relationships between different classes of modules are described.

## Abstract

Persistence modules are a central algebraic object arising in topological data analysis. The notion of interleaving provides a natural way to measure distances between persistence modules. We consider various classes of persistence modules, including many of those that have been previously studied, and describe the relationships between them. In the cases where these classes are sets, interleaving distance induces a topology. We undertake a systematic study the resulting topological spaces and their basic topological properties.