Definition 4.13.5 (Path Component). Let $X$ be a topological space and $A \subset X$ be path-connected, then there exists a unique path-connected set $C \supset A$ such that for any $C' \supset A$ path-connected, $C \supset C'$. The set $C$ is the path-component of $A$.
Proof. Let $C$ be the union of all path-connected sets containing $A$, then $C$ is path-connected by Proposition 4.13.4 and the maximum path-connected set containing $A$ by definition.$\square$