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

Theorem

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

Let the components of $T$ be open sets.

Then:

$S$ is a union of open connected sets of $T$.

Proof

Let the components of $T$ be open.

By definition, the components of $T$ are a partition of $S$.

Hence $S$ is the union of the open components of $T$.

Since a component is a maximal connected set by definition, then $S$ is a union of open connected sets of $T$.

$\blacksquare$