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In the Stone-Čech compactification of a discrete space, the closure of each subset of $X$ is clopen, because its indicator extends to a continuous zero-one-valued function. For disjoint closed sets $A,B\subseteq \beta X$, choose $u=0$ on $A$ and $u=1$ on $B$ by Urysohn's lemma. Then $U=\overline{\{x\in X:u(x)<1/2\}}$ and $V=\beta X\setminus U$ are disjoint clopen sets containing $A$ and $B$, respectively. Density proves these containments.

The Stone-Cech compactification $\beta \mathbb{N}$ can be identified with the space of ultrafilters on $\mathbb{N}$. Its basic clopen sets are $\bar{A}=\{\mathcal{U}:A\in \mathcal{U}\}$.