# Components are Open iff Union of Open Connected Sets

## Theorem

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

The following are equivalent::

- The components of $T$ are open.

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

## Proof

## Also see

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