# Completely Normal iff Every Subspace is Normal

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## Theorem

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

Then $T$ is a completely normal space if and only if every subspace of $T$ is normal.

## Proof

From the definitions, we have that:

- $T$ is a completely normal space if and only if:
- $\struct {S, \tau}$ is a $T_5$ space
- $\struct {S, \tau}$ is a $T_1$ (Fréchet) space.

- $T$ is a normal space if and only if:
- $\struct {S, \tau}$ is a $T_4$ space
- $\struct {S, \tau}$ is a $T_1$ (Fréchet) space.

From $T_1$ Property is Hereditary, any subspace of a $T_1$ space is also a $T_1$ space.

Then we have that a space is $T_5$ iff every subspace is $T_4$.

Hence the result.

$\blacksquare$

## 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: Functions, Products, and Subspaces