characterization of T2 spaces

Proposition 1.

[1, 2] Suppose X is a topological spaceMathworldPlanetmath. Then X is a T2 space ( if and only if for all xX, we have

{x} = {AAX𝑐𝑙𝑜𝑠𝑒𝑑, open setUsuch thatxUA}. (1)

By manipulating the definition using de Morgan’s laws, the claim can be rewritten as

{x}={VVXopen, open setUsuch thatxUV}.

Suppose y{x}. As X is a T2 space, there are open sets U,V such that xU,yV, and UV=. Thus, the inclusion from left to right holds. On the other hand, suppose yV for some open V such that {x}V. Then


and the claim follows. ∎


If we adopt the notation that a neighborhoodMathworldPlanetmathPlanetmath of x is any set containing an open set containing x, then the equation 1 can be written as

{x} = {AAXis a closed neighborhood of x}.


  • 1 L.A. Steen, J.A.Seebach, Jr., Counterexamples in topology, Holt, Rinehart and Winston, Inc., 1970.
  • 2 N. Bourbaki, General Topology, Part 1, Addison-Wesley Publishing Company, 1966.
Title characterizationMathworldPlanetmath of T2 spaces
Canonical name CharacterizationOfT2Spaces
Date of creation 2013-03-22 14:41:47
Last modified on 2013-03-22 14:41:47
Owner matte (1858)
Last modified by matte (1858)
Numerical id 7
Author matte (1858)
Entry type TheoremMathworldPlanetmath
Classification msc 54D10
Related topic LocallyCompactHausdorffSpace