Equivalence of Definitions of Component/Lemma 1

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

Let $x \in T$.

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

Let $\ds C = \bigcup \CC_x$

Then:
 * $C$ is connected in $T$ and $C \in \CC_x$.

Proof
From Singleton is Connected in Topological Space, $\set{x}$ is a connected set of $T$ containing $x$.

It follows that $x \in C$.

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

Hence $C \in \CC_x$.