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Aug 4, 2015 at 10:29 comment added Simon Henry You actually don't need $X$ and $Y$ to be Hausdorff to show that the homeomorphism cannot be induced by $\phi$. You only need them to have at least three points (otherwise the various topology on a two point spaces are all counterexample).
Aug 4, 2015 at 10:24 vote accept Dominic van der Zypen
Aug 4, 2015 at 10:16 history answered Adam Przeździecki CC BY-SA 3.0