 Compact Hausdorff topology on ultrafilters

For $A\subseteq \mathbb{N}$, the basic set $\hat{A}=\{U:A\in U\}$ 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.