# Perfectly Normal Space is Completely Normal Space

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## Theorem

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

Then $T$ is also a completely normal space.

## Proof

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

From the definition:

- $T$ is a perfectly $T_4$ space
- $T$ is a $T_1$ (Fréchet) space.

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## Sources

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $2$: Separation Axioms: Additional Separation Properties