Partition Topology is T4

Corollary to Partition Topology is T5

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

Let $T = \struct {S, \tau}$ be the partition space whose basis is $\PP$.

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

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $5$. Partition Topology: $2$