Quasicomponent is Intersection of Clopen Sets

Theorem
Let $T = \left({X, \tau}\right)$ be a topological space.

Let $p \in X$.

Then the quasicomponent containing $p$ equals the intersection of all sets which are both open and closed which contain $p$.

Proof
Let $C$ be the quasicomponent of $p$.

Let $Q$ be the set of clopen sets containing $p$ that are not equal to $X$.

First let $x \in C$.

Then for each $U \in Q$, $p \in U$.

Furthermore, since $U$ is clopen, $V$ is open, so $U$ and $V$ separate $X$.

Thus by the definition of quasicomponent, $x \in U$.

As this holds for all such $U$, $x \in \bigcap Q$.

As this holds for all such $x$, $C \subseteq Q$.

Now instead let $x \in \bigcap Q$.

Let $U$ and $V$ separate $X$, and let $p \in U$.

Then $U \in Q$.

Thus $x \in U$.

Thus $x$ lies on the same side as $p$ of any separation of $X$, so $x \in C$.