Equivalence of Definitions of Path Component

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

Let $x \in T$.

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
Let $C’$ be the equivalence class containing $x$ of the equivalence relation $\sim$ defined by:
 * $y \sim z$ $y$ and $z$ are connected in $T$.

Equivalence Class equals Union of Path-Connected Sets
It needs to be shown that $C = C’$.

Also see

 * Path-Connectedness is Equivalence Relation