Equivalence of Definitions of Path Component/Lemma 1

Theorem

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

Let $x \in T$.

Let $\CC_x = \leftset {A \subseteq S : x \in A \land A}$ is path-connected in $\rightset T$

Let $C = \bigcup \CC_x$

Then:

$C$ is path-connected in $T$ and $C \in \CC_x$.

Proof

From Point is Path-Connected to Itself, $\set x$ is a path-connected subset 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 subset of $T$.

Hence $C \in \CC_x$.

$\blacksquare$