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 17:50, 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