Definition:Completely Normal Space

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

$\left({S, \tau}\right)$ is a completely normal space iff:
 * $\left({S, \tau}\right)$ is a $T_5$ space
 * $\left({S, \tau}\right)$ is a $T_1$ (Fréchet) space.

That is:


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


 * $\forall x, y \in S$, both:
 * $\exists U \in \tau: x \in U, y \notin U$
 * $\exists V \in \tau: y \in V, x \notin V$