Quasicomponent of Compact Hausdorff Space is Connected

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Theorem

Let $\left({X, \tau}\right)$ be a compact Hausdorff space.

Let $C$ be a quasicomponent of $\left({X, \tau}\right)$.


Then $C$ is connected.


Proof

Let $p \in C$.

Aiming for a contradiction, suppose that $C$ is not connected.

Therefore, by definition of connected, there exist disjoint closed sets $A, B$ of $\left({X, \tau}\right)$ such that $C = A \cup B$.

By Compact Hausdorff Space is T4, there exist disjoint open sets $U, V$ of $\left({X, \tau}\right)$ such that $U \supseteq A$ and $V \supseteq B$.

By Quasicomponent is Intersection of Clopen Sets, $C$ is the intersection of all clopen sets of $\left({X, \tau}\right)$ containing $p$.

Since $U$ and $V$ are open, $X \setminus \left({U \cup V}\right)$ is closed.

Hence $X \setminus \left({U \cup V}\right)$ is compact.


Let $S$ be the set of clopen sets of $\left({X, \tau}\right)$ containing $p$.

Let $S'$ be the set of complements relative to $\left({X, \tau}\right)$ of elements of $S$.

Then $S'$ is an open cover of $X \setminus \left({U \cup V}\right)$.


Thus by compactness has a finite subcover $T'$.


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


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


Furthermore, since $T$ is a finite set of clopen sets of $\left({X, \tau}\right)$]], $\bigcap T$ is clopen.

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

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 of $\left({X, \tau}\right)$.


It follows that $C$ is connected.

$\blacksquare$