  Persistence modules are a central algebraic object arising in topological
data analysis. The notion of interleaving provides a natural way to measure
distances between persistence modules. We consider various classes of
persistence modules, including many of those that have been previously studied,
and describe the relationships between them. In the cases where these classes
are sets, interleaving distance induces a topology. We undertake a systematic
study the resulting topological spaces and their basic topological properties.
