Completely Normal iff Every Subspace is Normal

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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