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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?

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    $\begingroup$ No.${}{}{}{}{}$ $\endgroup$ – user98602 Aug 24 '16 at 18:32
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    $\begingroup$ @MikeMiller Ok, why? $\endgroup$ – Argon Aug 24 '16 at 18:33
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    $\begingroup$ Related: Is there a general way to tell whether two topological spaces are homeomorphic?. $\endgroup$ – Watson Aug 24 '16 at 18:33
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    $\begingroup$ Because if there was, then topology would be boring. If you want a rigorous answer, you should give a rigorous definition to "topological property". $\endgroup$ – user98602 Aug 24 '16 at 18:36
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    $\begingroup$ If a property means "A sentence in the language of topology", where we say that $X$ has the property if $\phi(X)$ is true, then your question is clearly false: this partitions the set of topological spaces into $2^n$ equivalence classes, and there are certainly more than $2^n$ non-homeomorphic spaces. But I suspect you intend to consider eg, cardinality, a "topological property", as opposed to "the cardinality is the same as that of $\Bbb R$". $\endgroup$ – user98602 Aug 24 '16 at 18:49
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Of course, there are trivial answers. As in the linked question, you want nontrivial answers. There is absolutely no prospect of such a list of techniques being developed by humans: in fact, the few techniques that can do anything to differentiate homotopy equivalent but non-homeomorphic spaces are generally limited to very special classes of space.

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