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

Theorem
Let $T = \struct {S, \tau}$ be a topological space.

Let $S$ be a union of open connected sets of $T$.

Then:
 * The components of $T$ are open sets.

Proof
Let $S = \ds \bigcup \set {U \subseteq S : U \in \tau \text { and } U \text { is connected} }$.

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

Lemma

 * For any connected set $U$ then:

Then:

Hence $C$ is the union of open sets.

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

The result follows.