  In this paper we study the properties of the homology of different geometric
filtered complexes (such as Vietoris-Rips, Cech and witness complexes) built on
top of precompact spaces. Using recent developments in the theory of
topological persistence we provide simple and natural proofs of the stability
of the persistent homology of such complexes with respect to the
Gromov--Hausdorff distance. We also exhibit a few noteworthy properties of the
homology of the Rips and Cech complexes built on top of compact spaces.
