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. Let the path components of $T$ be open sets. Then:
 * $S$ is a union of open path-connected sets of $T$.

Proof
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 $T$