# Condition for Closed Extension Space to be T4 Space

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## Theorem

Let $T = \left({S, \tau}\right)$ be a topological space.

Let $T^*_p = \left({S^*_p, \tau^*_p}\right)$ be the closed extension space of $T$.

Then $T^*_p$ is a $T_4$ space if and only if $T$ is a $T_4$ space vacuously, and $T^*_p$ in this case is also a $T_4$ space vacuously.

## Proof

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{II}: \ 12: \ 21$