  This paper presents explicit assumptions for the existence of interleaving
homotopy equivalences of both Vietoris-Rips and Lesnick complexes associated to
an inclusion of data sets. Consequences of these assumptions are investigated
on the space level, and for corresponding hierarchies of clusters and their
sub-posets of branch points. Hierarchy posets and branch point posets admit a
calculus of least upper bounds, which is used to show that the map of branch
points associated to the inclusion of data sets is a controlled homotopy
equivalence.
