Normal space

Definition

A topological space $X$ is normal when the property that follows holds. For all disjoint closed sets $A, B \subseteq X$, there are open sets $U$ and $V$ with $A \subseteq U$, $B \subseteq V$ and $U \cap V = \varnothing$.

Remarks

Names vary. Some authors call a space T4 when it is normal and T1. Such a space is regular and Hausdorff. Each metrizable space is normal, and each compact Hausdorff space is normal. A product of two normal spaces need not be normal. The Sorgenfrey plane is an example. Urysohn's lemma characterises normal spaces by continuous functions.

Explanations

L1 human

In a normal space, two closed sets that do not meet can be put into two open sets that do not meet. This is a strong condition, but it holds for all metric spaces.