Definition:Completely Normal Space

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

$\left({X, \vartheta}\right)$ is a completely normal space iff:


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

That is:
 * $\left({X, \vartheta}\right)$ is completely normal when for any two separated sets $A, B \subseteq X$ there exist disjoint open sets $U, V \in \vartheta$ containing $A$ and $B$ respectively.

Equivalent Definitions

 * $\left({X, \vartheta}\right)$ is completely normal iff each subset $Y$ contains a closed neighborhood of each $A \subseteq Y^\circ$ where $A^- \subseteq Y$.

In the above, $Y^\circ$ denotes the interior of $Y$ and $Y^-$ denotes the closure of $Y$.

This is proved in Equivalent Definitions for Completely Normal Space.

Variants of Name
Earlier (pre-1970) treatment of this subject tends to refer to this as a $T_5$ space, and what we define as a $T_4$ space as a completely normal space.