# Basis for Partition Topology

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## Theorem

Let $S$ be a set.

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

Let $\tau$ be the partition topology on $S$ defined as:

Then $\mathcal P$ forms a basis of $\tau$.

## Proof

Checking the criteria for $\mathcal P$ to be a synthetic basis for $\tau$:

We have that $\displaystyle S = \bigcup \mathcal P$ from the definition of a partition.

Therefore, $\displaystyle S \subseteq \bigcup \mathcal P$ and $\mathcal P$ is a cover for $S$.

Next, let $B_1, B_2 \in \mathcal P$.

Then as $\mathcal P$ is a partition of $S$, we have that $B_1 \cap B_2 = \varnothing$.

But from Union of Empty Set we have that $\varnothing$ is the (vacuous) union of sets of $\mathcal P$.

Hence $\mathcal P$ is a synthetic basis for $\tau$.

$\blacksquare$

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{II}: \ 5$