Strong topology on the set of persistence diagrams

TL;DR

This paper introduces a strong topology on the set of persistence diagrams, describes its properties, and proves that this space has infinite asymptotic dimension in the Gromov sense.

Contribution

It defines a new strong topology on persistence diagrams and analyzes its topological properties, including the infinite asymptotic dimension.

Findings

The space of persistence diagrams with the strong topology is characterized.

The space of persistence diagrams with the bottleneck metric has infinite asymptotic dimension.

The topology is described as a countable direct limit of bounded subsets.

Abstract

We endow the set of persistence diagrams with the strong topology (the topology of countable direct limit of increasing sequence of bounded subsets considered in the bottleneck distance). The topology of the obtained space is described. Also, we prove that the space of persistence diagrams with the bottleneck metric has infinite asymptotic dimension in the sense of Gromov.