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