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:

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


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