# Demonstrate Topological Spaces are Homeomorphic

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?

• No.${}{}{}{}{}$
– user98602
Aug 24, 2016 at 18:32
• @MikeMiller Ok, why? Aug 24, 2016 at 18:33
• Aug 24, 2016 at 18:33
• Because if there was, then topology would be boring. If you want a rigorous answer, you should give a rigorous definition to "topological property".
– user98602
Aug 24, 2016 at 18:36
• 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 $\Bbb R$".
– user98602
Aug 24, 2016 at 18:49