= Compact Hausdorff topology on ultrafilters
For $A\subseteq\mathbb N$, the basic set $\widehat A=\{U:A\in U\}$ in the <Stone-Čech compactification of the natural numbers> is a <clopen set>, since its complement is $\widehat{\mathbb N\setminus A}$. Distinct <ultrafilters> are separated by complementary basic sets. A basic cover with no finite subcover would give complements with the <finite intersection property>, whose generated <filter on a set> extends to an <ultrafilter> missing the entire cover. Thus the space is a <compact Hausdorff space>.
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