Partition Topology is Topology

Theorem

Let $S$ be a set.

Let $\PP$ 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 $\PP$

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

Proof

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

The result follows.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $5$. Partition Topology