Equivalence of Definitions of Path Component/Union of Path-Connected Sets is Maximal Path-Connected Set

Theorem
Let $T = \struct {S, \tau}$ be a topological space.

Let $x \in T$.

Let $\mathcal C_x = \left\{ {A \subseteq S : x \in A \land A } \right.$ is path-connected in $\left. {T} \right\}$

Let $C = \bigcup \mathcal C_x$

Then $C$ is a maximal path-connected set of $T$.

Lemma
Let $\tilde C$ be any path-connected set such that:
 * $C \subseteq \tilde C$

Then $x \in \tilde C$.

Hence $\tilde C \in \mathcal C_x$.

From Set is Subset of Union,
 * $\tilde C \subseteq C$.

Hence $\tilde C = C$.

It follows that $C$ is a maximal path-connected set of $T$ by definition.

Also see

 * Maximal Path-Connected Set is Union of Path-Connected Sets