= Šura-Bura component lemma
{c}
In a compact Hausdorff space, the <quasicomponent> $Q_x$ is connected and equals the <connected component> of $x$. If $Q_x=A\cup B$ were a nontrivial separation, disjoint open neighborhoods $U,V$ of its compact pieces leave a compact complement missing $Q_x$. Finitely many clopen neighborhoods of $x$ intersect to a clopen $N$ containing $Q_x$ inside $U\cup V$. The piece $N\cap U$ is then clopen in the whole space, contains $x$, and misses the nonempty other piece, contradicting the definition of $Q_x$. This also proves that any neighborhood of a component contains a clopen neighborhood of that component, by compactness of its complement.
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