Why is the classification of topological spaces up to homeomorphism impossible or undesirable?
Solution 1:
Often when people talk about "classifying" they think about having an algorithm (or a method) such that it takes a pair of spaces as an input and returns an answer to the question "are they homeomorphic/homotopic/diffeormorphic, etc?" as an output. All of that in a finite number of steps (see: decidability).
Such algorithm cannot exist because in particular we would be able to restrict the algorithm to manifolds. And in manifold case it is know that the problem is at least as hard as the word problem. And the word problem is known to have no solution.
Also IMO it would be very desirable if possible. I mean, what exactly would be a disadvantage of having such method?