Singleton Partition yields Indiscrete Topology

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $S$ be a set which is not empty.

Let $\mathcal P$ be the (trivial) singleton partition $\set S$ on $S$.


Then the partition topology on $\mathcal P$ is the indiscrete topology.


Proof

By definition, the partition topology on $\mathcal P$ is the set of all unions from $\mathcal P$.

This is (trivially, and from Union of Empty Set) $\set {\O, S}$ which is the indiscrete topology on $S$ by definition.

$\blacksquare$


Sources