= Clopen separation in discrete Stone-Čech compactifications
{title2=$\beta X=U\mathbin{\dot\cup}V$}
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.
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