# Partition Topology is Topology

## Theorem

Let $S$ be a set.

Let $\mathcal P$ be a partition of $S$.

Let $\tau$ be the set of subsets of $S$ defined as:

$a \in \tau \iff a$ is the union of sets of $\mathcal P$

Then $\tau$ is a topology on $S$.

## Proof

From Basis for Partition Topology, we have that $\mathcal P$ is a basis for the partition topology.

The result follows.

$\blacksquare$