  In this paper, we study the stability of commonly used filtration functions
in topological data analysis under small pertubations of the underlying
nonrandom point cloud. Relying on these stability results, we then develop a
test procedure to detect and determine structural breaks in a sequence of
topological data objects obtained from weakly dependent data. The proposed
method applies for instance to statistics of persistence diagrams of
$\mathbb{R}^d$-valued Bernoulli shift systems under the \v{C}ech or
Vietoris-Rips filtration.
