\begin{definition}[Definition:Normal Space]
Let $T = \struct {S, \tau}$ be a topological space.
$\struct {S, \tau}$ is a '''normal space''' {{iff}}:
:$\struct {S, \tau}$ is a $T_4$ space
:$\struct {S, \tau}$ is a $T_1$ (Fréchet) space.
That is:
:$\forall A, B \in \map \complement \tau, A \cap B = \O: \exists U, V \in \tau: A \subseteq U, B \subseteq V, U \cap V = \O$ 
:$\forall x, y \in S$, ''both'':
::$\exists U \in \tau: x \in U, y \notin U$
::$\exists V \in \tau: y \in V, x \notin V$
{{SeparationAxiomNameVariants||T4|$T_4$||normal}}
This space is also referred to as '''normal Hausdorff'''.
\end{definition}