# Does compact Hausdorff imply Polish? [duplicate]

Possible Duplicate:
Is a compact Hausdorff space metrizable? Maybe even complete?

We know that a second countable locally compact Hausdorff space is a Polish space. Does compact Hausdorff also imply Polish?

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## marked as duplicate by t.b., Martin Sleziak, Asaf Karagila, Michael Greinecker♦, Leonid Kovalev Aug 14 '12 at 18:22

No: $\omega_1+1$ with the order topology is compact and Hausdorff but not even first countable, let alone metrizable.
Added: $\beta\Bbb N$, the Čech-Stone compactification of the natural numbers with the discrete topology, is another example, and it’s even separable.