completely Hausdorff

Definition 1.

[1] Let (X,τ) be a topological spaceMathworldPlanetmath. Suppose that for any two different points x,yX,xy, we can find two disjoint neighborhoodsMathworldPlanetmathPlanetmath


such that their closuresMathworldPlanetmathPlanetmath are also disjoint:


Then we say that (X,τ) is a completely Hausdorff space or a T212 space.


A synonym for functionally Hausdorff space is Urysohn space [1]. Unfortunately, the definition of completely HausdorffPlanetmathPlanetmath and T212 are not as standard as one would like since. For example, the term completely Hausdorff space is also used to mean a functionally Hausdorff space (e.g. [2]). Nevertheless, in the present convention, we have the implicationMathworldPlanetmath:

functionally Hausdorffcompletely HausdorffT2=Hausdorff,

which suggests why the T212 name have been used to denote both completely Hausdorff spaces and functionally Hausdorff spaces.


  • 1 L.A. Steen, J.A.Seebach, Jr., Counterexamples in topology, Holt, Rinehart and Winston, Inc., 1970.
  • 2 S. Willard, General Topology, Addison-Wesley Publishing Company, 1970.
Title completely Hausdorff
Canonical name CompletelyHausdorff
Date of creation 2013-03-22 14:16:03
Last modified on 2013-03-22 14:16:03
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 15
Author PrimeFan (13766)
Entry type Definition
Classification msc 54D10
Synonym completely Hausdorff space
Synonym T212
Synonym Urysohn space
Related topic HausdorffSpaceNotCompletelyHausdorff