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

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

Let $x \in T$.

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

Let $C = \bigcup \mathcal C_x$

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

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

Then $C = C’$.

Proof
The result follows.

Also see

 * Connectedness of Points is Equivalence Relation