regular space

Definition 1.

A topological spaceMathworldPlanetmath is a regular spaceMathworldPlanetmathPlanetmath if it is both a T0 space ( and a T3 space (

Example. Consider the set with the topologyMathworldPlanetmath σ generated by the basis

β={U=V-C:V is open with the standard topology and C is (infinite) numerable}.

Since is numerable and open, the set of irrational numbers - is open and therefore is closed. It can be shown that - is an open set with this topology and is closed.

Take any irrational number x. Any open set V containing all must contain also x, so the regular space property cannot be satisfied. Therefore, (,σ) is not a regular space.


In topology, the terminology for separation axiomsMathworldPlanetmathPlanetmath is not standard. Therefore there are also other meanings of regular. In some references (e.g. [2]) the meanings of regular and T3 is exchanged. That is, T3 is a stronger property than regular.


  • 1 L.A. Steen, J.A.Seebach, Jr., Counterexamples in topology, Holt, Rinehart and Winston, Inc., 1970.
  • 2 J.L. Kelley, General Topology, D. van Nostrand Company, Inc., 1955.
Title regular space
Canonical name RegularSpace
Date of creation 2013-03-22 12:18:21
Last modified on 2013-03-22 12:18:21
Owner drini (3)
Last modified by drini (3)
Numerical id 11
Author drini (3)
Entry type Definition
Classification msc 54D10
Synonym regular
Related topic SeparationAxioms
Related topic T0Space
Related topic T2Space
Related topic T3Space
Related topic HausdorffSpaceNotCompletelyHausdorff
Related topic T1Space