Gromov-Hausdorff space is a metric space of compact metric spaces and is useful in many areas of geometry.  Motivated by Cheeger's thesis, there are a number of results proving that for many pre-compact sets in GH space, there are only finitely many homeomorphism types of manifolds.  I will explain some work with Sasha Dranishnikov and Steve Ferry that shows infinite dimensional phenomena arise in certain effective versions of this result.  This indirectly leads to a certain kind of metric-topological rigidity that holds for all manifolds whose fundamental groups are lattices in real Lie groups (or are word-hyperbolic), but not shared by all those whose fundamental groups are linear.