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

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

Let $x \in T$. Let $\tilde C$ be a maximal path-connected set of $T$ that contains $x$.

Then:
 * $\tilde C = \bigcup \set {A \subseteq S : x \in A \land A \text{ is path-connected in } T}$.

Proof
Let $\mathcal C_x = \set {A \subseteq S : x \in A \land A \text{ is path-connected in } T}$

Let $C = \bigcup \mathcal C_x$

Lemma
By definition, $\tilde C \in \mathcal C_x$.

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

By maximality of $\tilde C$ then $\tilde C = C$

Also see

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