 Šura-Bura lemma

In a compact Hausdorff space, each connected component is its quasi-component and has a neighborhood basis of clopen sets. Let $Q$ be the intersection of clopen neighborhoods of $x$. It is compact. If it splits into two nonempty compact sets, normality separates them by disjoint open neighborhoods. The finite intersection characterization of compactness gives a finite intersection of clopen neighborhoods lying in their union. Its intersection with the neighborhood containing $x$ is then clopen and excludes part of $Q$, a contradiction. Thus $Q$ is connected and equals the component. Applying the same finite-intersection argument to any neighborhood of $Q$ proves the neighborhood-basis assertion. ^[ref].