Separation axioms
From Citizendium, the Citizens' Compendium
In topology, separation axioms describe classes of topological space according to how well the open sets of the topology distinguish between distinct points.
Contents
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 . 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 .
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 if T0 and T3
- completely regular if T0 and T3½
- Tychonoff is completely regular and T1
- normal if T0 and T4
- completely normal if T1 and T5
- perfectly normal if normal and every closed set is a G_{δ}
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'.
References
- Steen, Lynn Arthur & J. Arthur Jr. Seebach (1978), Counterexamples in Topology, Berlin, New York: Springer-Verlag, ISBN 0-387-90312-7