Separation axioms: Difference between revisions
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In [[topology]], '''separation axioms''' describe classes of [[topological space]] according to how well the [[open set]]s of the topology distinguish between distinct points | In [[topology]], '''separation axioms''' describe classes of [[topological space]] according to how well the [[open set]]s of the topology distinguish between distinct points. | ||
Revision as of 16:50, 31 October 2008
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 . 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 .
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