Path Components are Open iff Union of Open Path-Connected Sets

Theorem
Let $T = \struct {S, \tau}$ 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