Separation axioms

In topology, separation axioms describe classes of topological space according to how well the open sets of the topology distinguish between distinct points

Terminology
A neighbourhood of a point x in a topological space X is a set N such that x is in the interior of N; that is, there is an open set U such that $$x \in U \subseteq N$$. A neighbourhood of a set A in X is a set N such that A is contained in the interior of N; that is, there is an open set U such that $$A \subseteq U \subseteq N$$.

Properties
A topological space X is
 * Hausdorff if any two distinct points have disjoint neighbourhoods
 * normal if a closed set A and a point x not in A have disjoint neighbourhoods
 * regular if disjoint closed sets have disjoint neighbourhoods