T4 Property Preserved in Closed Subspace/Corollary

Theorem
Let $T = \struct {S, \tau}$ be a topological space.

Let $T_K$ be a subspace of $T$ such that $K$ is closed in $T$.

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

That is, the property of being a normal space is weakly hereditary.

Proof
From the definition, $T = \struct {S, \tau}$ is a normal space :
 * $\struct {S, \tau}$ is a $T_4$ space
 * $\struct {S, \tau}$ 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 T4 Property Preserved in Closed Subspace, any closed subspace of a $T_4$ space is also a $T_4$ space.

Hence the result.