Completely Normal Space is Normal Space

Theorem
Let $\left({X, \vartheta}\right)$ be a completely normal space.

Then $\left({X, \vartheta}\right)$ is also a normal space.

Proof
Let $\left({X, \vartheta}\right)$ be a completely normal space.

From the definition of completely normal space:


 * $\forall A, B \subseteq X, A^- \cap B = A \cap B^- = \varnothing: \exists U, V \in \vartheta: A \subseteq U, B \subseteq V, U \cap V = \varnothing$

where $A^-$ is the closure of $A$ in $T$.

Let $C, D \subseteq X$ be disjoint sets which are closed in $T$.

Thus $C, D \in \complement \left({\vartheta}\right)$ from the definition of closed set.

From Closure is Closed, we have:
 * $C^- = C, D^- = D$

and so it follows from $C \cap D = \varnothing$ that:
 * $C^- \cap D = C \cap D^- = \varnothing$

Thus we have, from the definition of completely normal space:


 * $\forall C, D \in \complement \left({\vartheta}\right), C \cap D = \varnothing: \exists U, V \in \vartheta: C \subseteq U, D \subseteq V$

which is precisely the definition of a normal space.