Let $f:X \rightarrow Y$ be a bijective, continuous map between two toplogical spaces. Does X being compact and Hausdorff imply that $f$ must be a homeomorphism? I think it doesn't but I can not find an example.


Let $X$ be your favourite compact Hausdorff space with at least $2$ points, let $Y$ be $X$ with the trivial topology, consider the identity as $f$.

  • $\begingroup$ I think you should specify what you mean by trivial: is it the discrete topology or the anti-discrete topology? This clearly cannot be true for the discrete topology since $f$ itself need not be continuous. $\endgroup$ – weierstrash Jun 26 '20 at 16:27
  • $\begingroup$ @PankajTiwari trivial is the common name for the topology with only two open sets even though some authors use indiscrete or other names $\endgroup$ – Alessandro Codenotti Jun 26 '20 at 16:29
  • $\begingroup$ I thought the OP might not be familiar with the terminology so might end up getting confused. Anyways you're answer is of course correct so +1! $\endgroup$ – weierstrash Jun 26 '20 at 16:31
  • $\begingroup$ Also many people use the Hausdorff condition as an axiom for a topological space so one has to double check what defintions they're using. $\endgroup$ – weierstrash Jun 26 '20 at 16:33

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