# Metrics on sets of interval partitions with diversity

## TL;DR

This paper introduces a new metric on sets of interval partitions with Lebesgue-null complements, incorporating diversity measures, and proves the resulting space is Lusin with continuous diversity evolution.

## Contribution

It develops a complete metric for interval partitions with diversity, linking topology with diversity evolution, and establishes the space as Lusin.

## Key findings

- The metric induces the same topology as Hausdorff distance.
- The space of partitions with diversity is Lusin.
- Path-continuity implies continuous diversity evolution.

## Abstract

We first consider interval partitions whose complements are Lebesgue-null and introduce a complete metric that induces the same topology as the Hausdorff distance (between complements). This is done using correspondences between intervals. Further restricting to interval partitions with alpha-diversity, we then adjust the metric to incorporate diversities. We show that this second metric space is Lusin. An important feature of this topology is that path-continuity in this topology implies the continuous evolution of diversities. This is important in related work on tree-valued stochastic processes where diversities are branch lengths.