Path Components are Open iff 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 $T$.

Also see

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