Partition Topology is T3

Corollary to Partition Topology is T3 1/2

Let $S$ be a set and let $\PP$ be a partition on $S$.

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

Then:

$T$ is a $T_3$ space.

Proof

We have that the Partition Topology is $T_{3 \frac 1 2}$.

We also have that a $T_{3 \frac 1 2}$ Space is $T_3$ 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: $2$