# Open Extension Topology is T4

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## Theorem

Let $T = \struct {S, \tau}$ be a topological space.

Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the open extension space of $T$.

Then $T^*_{\bar p}$ is a $T_4$ space.

## Proof

We have that an Open Extension Space is Ultraconnected.

That means none of its closed sets are disjoint.

Hence, vacuously, any two of its disjoint closed subsets are separated by neighborhoods.

The result follows by definition of $T_4$ space.

$\blacksquare$

## Sources

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $16$. Open Extension Topology: $9$