I am aware of long lists of topological properties that are invariant under homeomorphism. One can prove that two spaces are not homeomorphic if they don't agree on a certain property (e.g. one space is Hausdorff and another is not). However, finding that two spaces agree on all these properties does not indicate that they are homeomorphic, just that they may be homeomorphic.
Can there exist a finite, exhaustive list of topological properties that prove with certainty that two spaces are homeomorphic?