Separation axioms

In topology, separation axioms describe classes of topological spaces 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 $$A \subseteq U \subseteq N$$.

Subsets U and V are separated in X if U is disjoint from the closure of V and V is disjoint from the closure of U.

A Urysohn function for subsets A and B of X is a continuous function f from X to the real unit interval such that f is 0 on A and 1 on B.

Axioms
A topological space X is
 * T0 if for any two distinct points there is an open set which contains just one
 * T1 if for any two points x, y there are open sets U and V such that U contains x but not y, and V contains y but not x
 * T2 if any two distinct points have disjoint neighbourhoods
 * T2½ if distinct points have disjoint closed neighbourhoods
 * T3 if a closed set A and a point x not in A have disjoint neighbourhoods
 * T3½ if for any closed set A and point x not in A there is a Urysohn function for A and {x}
 * T4 if disjoint closed sets have disjoint neighbourhoods
 * T5 if separated sets have disjoint neighbourhoods


 * Hausdorff is a synonym for T2
 * completely Hausdorff is a synonym for T2½


 * regular means T0 and T3
 * completely regular means T0 and T3½
 * Tychonoff means completely regular and T1


 * normal means T0 and T4
 * completely normal means T1 and T5
 * perfectly normal means: normal and such that every closed set is a Gδ

Properties

 * A space is T1 if and only if each point (singleton) forms a closed set.
 * Urysohn's Lemma: if A and B are disjoint closed subsets of a T4 space X, there is a Urysohn function for A and B.