On C0-persistent homology and trees

In this paper we give a metric construction of a tree which correctly identifies connected components of superlevel sets of ℝ-valued continuous functions f on X and show that it is possible to retrieve the H0-persistent diagram from this tree. We revisit the notion of homological dimension previously introduced by Schweinhart and give some bounds for the latter in terms of the upper-box dimension of X, thereby partially answering a question of the same author. We prove a quantitative version of the Wasserstein stability theorem valid for regular enough X and α-H"older functions and discuss some applications of this theory to random fields and the topology of their superlevel sets.