Is there a general way to tell whether two topological spaces are homeomorphic?

Solution 1:

No, at least in the sense that there are no such functors $F$ which are substantially more approachable than the original problem. There are very few general techniques for distinguishing homotopy equivalent, non-homeomorphic spaces: see the following link for work in the last twelve years distinguishing some such pairs of 3-dimensional closed manifolds, which are a ludicrously tiny tip of the iceberg of arbitrary spaces. It's worth noting here, to give a sense of how far from possible a simple, complete homeomorphism invariant is, that even the homotopy equivalence problem runs into uncomputability, insofar as it's undecidable whether two spaces have the same fundamental group, given presentations of those groups. https://mathoverflow.net/questions/242748/list-of-invariants-that-distinguish-homotopy-equivalent-non-homeomorphic-spaces