Quasicomponent ($Q_x=\bigcap \{U:x\in U,U\text{ clopen}\}$)

The quasicomponent of a point is the intersection of all its clopen neighborhoods. Every connected set containing the point lies in its quasicomponent, since a clopen set cannot split a connected set. In a compact Hausdorff space, quasicomponents equal connected components by the Šura-Bura component lemma.

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.