Path Components are Open iff Union of Open Path-Connected Sets/Space is Union of Open Path-Connected Sets implies Path Components are Open

Theorem
Let $T = \left({S, \tau}\right)$ be a topological space. Let $S$ be the union of open path-connected sets of $T$.

Then:
 * The path components of $T$ are open sets.

Proof
Let $S = \bigcup \{ U \subseteq S : U \in \tau \text { and } U \text { is path-connected} \}$.

Let $C$ be a path component of $T$.

Lemma

 * For any path-connected set $U$ then:

Then:

Hence $C$ is the union of open sets.

By definition of a topology then $C$ is an open set.

The result follows.