Completely Normal iff Every Subspace is Normal

Theorem
Let $T = \displaystyle \left({X, \vartheta}\right)$ be a topological space.

Then $T$ is a $T_5$ space iff every subspace of $T$ is a $T_4$ space.

Corollary
$T$ is a completely normal space iff every subspace of $T$ is normal.

Proof of Corollary
From the definitions, we have that:


 * $T$ is a completely normal space iff:
 * $\left({X, \vartheta}\right)$ is a $T_5$ space
 * $\left({X, \vartheta}\right)$ is a $T_1$ (Fréchet) space.


 * $T$ is a normal space iff:
 * $\left({X, \vartheta}\right)$ is a $T_4$ space
 * $\left({X, \vartheta}\right)$ is a $T_1$ (Fréchet) space.

From Separation Properties Preserved in Subspace, any subspace of a $T_1$ space is also a $T_1$ space.

From the main result, $T$ is a $T_5$ space iff every subspace of $T$ is a $T_4$ space.

Hence the result.