Definition:Normal Space

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

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


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

That is, for any two disjoint closed sets $A, B \subseteq X$ there exist disjoint open sets $U, V \in \vartheta$ containing $A$ and $B$ respectively.

That is:
 * $\left({X, \vartheta}\right)$ is normal when any two disjoint closed subsets of $X$ are separated by neighborhoods.

Equivalent Definitions
$\left({X, \vartheta}\right)$ is normal iff each open set $U$ contains a closed neighborhood of each closed set contained in $U$.

This is proved in Equivalent Definitions for Normal Space.

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