Partition Topology is T4

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Corollary to Partition Topology is T5

Let $S$ be a set and let $\mathcal P$ be a partition on $S$ which is not the (trivial) partition of singletons.

Let $T = \left({S, \tau}\right)$ be the partition space whose basis is $\mathcal P$.


Then:

$T$ is a $T_4$ space.


Proof

We have that the Partition Topology is $T_5$.

We also have that a $T_5$ Space is $T_4$ Space.

The result follows.

$\blacksquare$


Sources