Filtrations induced by continuous functions

TL;DR

This paper investigates the relationship between filtrations in persistent homology induced by ordered sets and those induced by continuous functions, establishing conditions under which they are equivalent.

Contribution

It proves that, under stability and completeness assumptions, continuous functions can induce all relevant filtrations in persistent homology, extending to multi-dimensional cases.

Findings

Every compact, stable 1D filtration of a compact metric space is induced by a continuous function.

The result extends to multi-dimensional filtrations with completeness assumptions.

Counterexamples show the necessity of stability and completeness assumptions.

Abstract

In Persistent Homology and Topology, filtrations are usually given by introducing an ordered collection of sets or a continuous function from a topological space to $\R^n$. A natural question arises, whether these approaches are equivalent or not. In this paper we study this problem and prove that, while the answer to the previous question is negative in the general case, the approach by continuous functions is not restrictive with respect to the other, provided that some natural stability and completeness assumptions are made. In particular, we show that every compact and stable 1-dimensional filtration of a compact metric space is induced by a continuous function. Moreover, we extend the previous result to the case of multi-dimensional filtrations, requiring that our filtration is also complete. Three examples show that we cannot drop the assumptions about stability and completeness. Consequences of our results on the definition of a distance between filtrations are finally discussed.