Equivalence of Definitions of Path Component/Lemma 1

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 path-connected in $T$ and $C\in \mathcal C_x$.

Proof
From Point is Path-Connected to Itself, $\set{x}$ is a path-connected of $T$ containing $x$.

It follows that $x \in C$.

From Union of Path-Connected Sets with Common Point is Path-Connected, $C$ is a path-connected of $T$.

Hence $C \in \mathcal C_x$.