Filtrations induced by continuous functions

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. 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. Introduction The concept of filtration is the start point for Persistent Topology and Homology. Actually, the main goal of these theories is to examine the topological and homological changes that happen when we go through a family of spaces that is totally ordered with respect to inclusion [12]. In literature, filtrations are usually given in two ways. The former consists of explicitly introducing a nested collection of sets (usually carriers of simplicial complexes), the latter of giving a continuous function from a topological space to R or R (called a filtering function), whose sub-level sets represent the elements of the considered filtration (cf., e.g., [11, 15]). An example of these two types of filtrations is shown in Figure 1. The two considered methods have produced two different approaches to study the concept of persistence. A natural question arises, whether these approaches are equivalent or not. In our 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 some sense, this statement shows that the approach by continuous functions (and the related theoretical properties) can be used without loss of generality, and represents the main result of this paper. The interest in this investigation is mainly due to the desire of building a bridge between the two settings, which would ensure that results available in literature for the approach by functions are also valid for the other method. As examples of results that have been proved in one setting and that it would be desirable to apply to the other, we can cite [5] and [4], in which persistence diagrams in the 1dimensional and n-dimensional setting, respectively, are proved to be stable shape descriptors via the use of the associated filtering functions. Another example can be found in [6], where a Mayer-Vietoris formula involving the ranks of persistent homology groups of a space and its subspaces is obtained by defining a filtering 2010 Mathematics Subject Classification. Primary 54E45; Secondary 65D18, 68U05.