  Topological data analysis is an emerging field that applies the study of
topological invariants to data. Perhaps the simplest of these invariants is the
number of connected components or clusters. In this work, we explore a
topological framework for cluster analysis and show how it can be used as a
basis for explainability in unsupervised data analysis. Our main object of
study will be hierarchical data structures referred to as Topological
Hierarchical Decompositions (THDs). We give a number of examples of how
traditional clustering algorithms can be topologized, and provide preliminary
results on the THDs associated with Reeb graphs and the mapper algorithm. In
particular, we give a generalized construction of the mapper functor as a
pixelization of a cosheaf in order to generalize multiscale mapper.
