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Let $f:(X,\tau_1) \to(Y,\tau_2)$ be an injective and surjective continuous function. If $X$ is compact with respect to $\tau_1$ and $Y$ is Hausdorff with respect to $\tau_2$ then how can we show that $f$ is a Homeomorphism?

I know that every bijective bi-continuous mappings are homeomorphic. Here it is given that this mapping is bijective and continuous, how can I show that the inverse map is continuous?

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Hint: All you need to show is that $f$ is either open or closed to have that it is a homeomorphism.

Note that the continuous image of a compact set is compact; and that closed sets in a compact space are compact.

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  • $\begingroup$ But how can I use the fact that Y is Hausdorff to say that the map is either closed or open? $\endgroup$ – ccc Nov 4 '12 at 11:45
  • $\begingroup$ Compact sets in Hausdorff spaces are closed. $\endgroup$ – Asaf Karagila Nov 4 '12 at 12:07
  • $\begingroup$ thanks alot Asaf Karagila:) $\endgroup$ – ccc Nov 4 '12 at 12:47
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Let $A \in \tau_1$ be a closed. Since closed subset of a compact space is compact so it is compact. Now continuous image of a compact set is compact so $f(A)$ is compact. Finally by using the theorem - Every compact set of a Hausdorff space is closed we will get $f(A)$ a closed set . Hence mapping is closed and prove is done.

I hope this will help you.

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