# A Primer on Persistent Homology of Finite Metric Spaces

## TL;DR

This paper provides a concise introduction to persistent homology, a key concept in topological data analysis, explaining its construction, invariance, and stability for analyzing datasets across scales.

## Contribution

It offers a self-contained overview of persistent homology, emphasizing its foundational ideas and stability properties in data analysis.

## Key findings

- Persistent homology captures scale-dependent topological features.
- It is stable under data perturbations.
- Provides a foundational understanding for TDA applications.

## Abstract

TDA (topological data analysis) is a relatively new area of research related to importing classical ideas from topology into the realm of data analysis. Under the umbrella term TDA, there falls, in particular, the notion of persistent homology, which can be described in a nutshell, as the study of scale dependent homological invariants of datasets. In these notes, we provide a terse self contained description of the main ideas behind the construction of persistent homology as an invariant feature of datasets, and its stability to perturbations.