TITLE: Is a compact Hausdorff space metrizable? Maybe even complete?
QUESTION [5 upvotes]: Possible Duplicate:
A compact Hausdorff space that is not metrizable 

Is it true that every topological space $X$ that is Hausdorff and compact is also metrizable? Maybe even complete?
What is the relationship between the completeness and the compactness in a metric space ?
I'm not asking this question out of nowhere. The reason is that almost every theorem I have encountered lately about compact metric spaces could easily be generalized to compact Hausdorff space but also some theorems I encountered about complete metrics have a counterpart for compact Hausdorff spaces  (e.g. Baire's theorem).

REPLY [8 votes]: To address your second question in perhaps a simpler way than Daria's answer: every compact metric space is complete.  This is easy to prove using the Bolzano-Weierstrass theorem: any Cauchy sequence $\{x_n\}$ has a subsequence that converges to some point $x$.  Then use the triangle inequality to show that the original sequence $\{x_n\}$ must converge to $x$.
Of course the converse is false: there are complete metric spaces which are not compact.  $\mathbb{R}$, for example.