Components are Open iff Union of Open Connected Sets

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

Then the following are equivalent:
 * The components of $T$ are open.
 * $S$ is a union of open connected subsets of $S$ in $T$.

Also see

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