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 $A\subseteq U\subseteq N$.

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.

Properties

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
• 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
• normal if T0 and T3
• regular if T0 and T4
• completely normal if T1 and T5
• perfectly normal if normal and every closed set is a Gδ

References

• Steen, Lynn Arthur & J. Arthur Jr. Seebach (1978), Counterexamples in Topology, Berlin, New York: Springer-Verlag, ISBN 0-387-90312-7