Quasicomponent of Compact Hausdorff Space is Connected

Theorem
Let $(X, \tau)$ be a compact Hausdorff space.

Let $C$ be a quasicomponent of $X$.

Then $C$ is connected.

Proof
Let $p \in C$.

Suppose for the sake of contradiction that $A$ and $B$ are disjoint closed sets and $C = A \cup B$.

By Compact Hausdorff Space is T4, there are disjoint open sets $U \supseteq A$ and $V \supseteq B$.

By Quasicomponent is Intersection of Clopen Sets, $C$ is the intersection of all clopen sets containing $p$.

Since $U$ and $V$ are open, $X \setminus (U \cup V)$ is closed, hence compact.

Let $S$ be the set of clopen sets containing $p$.

Let $S'$ be the set of complements of elements of $S$.

Then $S'$ is an open cover of $X \setminus (U \cup V)$, and thus has a finite subcover $T'$.

Let $T$ be the set of complements of elements of $T'$.

Then $C \subseteq \bigcap T \subseteq U \cup V$.

Furthermore, since $T$ is a finite set of clopen sets, $\bigcap T$ is clopen.

Let $U' = U \cap \bigcap T$ and let $V' = V \cap \bigcap T$.

Then $C \subseteq U' \cup V' = T$.

Since $T$ is clopen, so is $U'$.

But $C$ contains points in $U'$ and points not in $U'$, contradicting the fact that $C$ is a quasicomponent.

Thus we conclude that $C$ is connected.