In this dissertation, we study the persistent homology obtained by the homology on the open Rips filtration of a compact metric space $(X,d)$. We show that every decrease in $0$-dimensional persistent homology and every increase in $1$-dimensional persistent homology is a consequence of local minima of the distance function $d$. When $d$ attains a local minimum for only finitely many pairs of points, we prove that each such change in the persistent homology is caused by a specific critical edge in the Rips complexes corresponding to a local minimum of the function $d$. The results include upper bounds for the rank of the $1$-dimensional persistent homology and a corresponding reconstruction theorem. From a computational point of view, it is noteworthy that there is a simple geometric criterion to identify those local minima of the distance function d that cause changes in the persistent homology. These results provide the first interpretation of the critical values of the persistent homology (obtained by Rips complexes) for general compact metric spaces.
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