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

Theorem
Let $T = \left({S, \tau}\right)$ be a topological space.




 * (1) $\quad$ The path components of $T$ are open.


 * (2) $\quad S$ is a union of open path-connected sets of $T$.

(1) implies (2)
Let the path components of $T$ be open.

By definition, the path components of $T$ are a partition of $S$.

Hence $S$ is the union of the open path components of $T$.

Since a path component is a maximal path-connected set by definition, then $S$ is a union of open path-connected sets of $T$

(2) implies (1)
Let $S = \bigcup \{ U \subseteq S : U \in \tau \text { and } U \text { is path-connected} \}$.

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

Then:

Hence $C$ is the union of open sets.

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

The result follows.

Also see

 * Components are Open iff Union of Open Connected Sets, an analogous result for components