Equivalence of Definitions of Path Component/Equivalence Class equals Union of Path-Connected Sets

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 $C = \bigcup \mathcal C_x$

Let $\sim$ be the equivalence relation defined by:
 * $y \sim z$ $y$ and $z$ are path-connected in $T$.

Let $C’$ be the equivalence class of $\sim$ containing $x$.

Then $C = C’$.

Proof
The result follows.

Also see

 * Path-Connectedness is Equivalence Relation