# A Distance Between Filtered Spaces Via Tripods

## TL;DR

This paper introduces a new intrinsic distance measure between filtered spaces that enhances stability analysis and constructs explicit geodesics, improving understanding of filtrations on different finite spaces.

## Contribution

It extends stability results to filtrations on different spaces without interleaving and constructs explicit geodesics to demonstrate the intrinsic nature of the distance.

## Key findings

- Established a new intrinsic distance measure for filtered spaces.
- Constructed explicit geodesics between filtered spaces.
- Strengthened stability results for filtrations.

## Abstract

We present a simplified treatment of stability of filtrations on finite spaces. Interestingly, we can lift the stability result for combinatorial filtrations from [CSEM06] to the case when two filtrations live on different spaces without directly invoking the concept of interleaving. We then prove that this distance is intrinsic by constructing explicit geodesics between any pair of filtered spaces. Finally we use this construction to obtain a strengthening of the stability result.