T4 Property Preserved in Closed Subspace/Corollary

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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 if and only if:

$\struct {S, \tau}$ 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.

$\blacksquare$


Sources