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).