# 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

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{II}: \ 5: \ 2$