# Completely Normal iff Every Subspace is Normal

Jump to navigation
Jump to search

## Theorem

Let $T = \displaystyle \left({S, \tau}\right)$ 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:
- $\left({S, \tau}\right)$ is a $T_5$ space
- $\left({S, \tau}\right)$ is a $T_1$ (Fréchet) space.

- $T$ is a normal space if and only if:
- $\left({S, \tau}\right)$ is a $T_4$ space
- $\left({S, \tau}\right)$ 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

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{I}: \ \S 2$: Functions, Products, and Subspaces