T4 Property Preserved in Closed Subspace

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

Let $T_K$ be a subspace of $T$ which is closed in $T$.

If $T$ is a $T_4$ space then $T_K$ is also a $T_4$ space.

Corollary
If $T$ is a normal space then $T_K$ is also a normal space.

Proof of Corollary
From the definition, $T$ 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, any closed subspace of a $T_4$ space is also a $T_4$ space.

Hence the result.