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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?

Argon
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    No.${}{}{}{}{}$ –  Aug 24 '16 at 18:32
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    @MikeMiller Ok, why? – Argon Aug 24 '16 at 18:33
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    Related: Is there a general way to tell whether two topological spaces are homeomorphic?. – Watson Aug 24 '16 at 18:33
  • There's a difference between, "does there currently exist" and "can there exist." – Mike Pierce Aug 24 '16 at 18:34
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    Because if there was, then topology would be boring. If you want a rigorous answer, you should give a rigorous definition to "topological property". –  Aug 24 '16 at 18:36
  • @MikeMiller According to Munkres, given a homeomorphism $f : X \to Y$, "any property of $X$ that is entirely expressed in terms of the topology of $X$ (that is, in terms of the open sets of $X$) yields, via the [homeomorphic] correspondence $f$, the corresponding property of the space $Y$. Such a property of $X$ is called a topological property of $X$." What additional rigour would you need? – Argon Aug 24 '16 at 18:44
  • What is a property? –  Aug 24 '16 at 18:47
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    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 $\Bbb R$". –  Aug 24 '16 at 18:49
  • "Cardinality" certainly counts as a topological property. So, apparently, must "the isomorphism class of the lattice of open subsets together with the inclusion relation between points and open subsets," that is, the homeomorphism type itself. So it doesn't seem possible to prove there's no finite list-this one's just useless. – Kevin Carlson Aug 24 '16 at 19:28

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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.

Kevin Carlson
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