Definition 4.12.4 (Connected Component). Let $X$ be a topological space and $A \subset X$ be connected, then there exists a unique connected set $C \supset A$ such that for any $C' \supset A$ connected, $C \supset C'$. The set $C$ is the connected component of $A$.
Proof. Let $C$ be the union of all connected sets that contain $A$, then $C$ is connected by Proposition 4.12.2, and is the maximum connected set containing $A$ by definition.$\square$