# Open Set in Partition Topology is Component

## Theorem

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

Then each of its open sets are components of $T$.

## Proof

Let the partition $\PP$ be a basis of $T$.

From Open Set in Partition Topology is also Closed, open sets are in fact clopen.

So the elements of $\PP$ are clopen.

The result follows from the definition of components.

$\blacksquare$