Completely Normal iff Every Subspace is Normal

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

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

Proof
From the definitions, we have that:


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


 * $T$ is a normal space :
 * $\left({S, \tau}\right)$ 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.