Hausdorff space

After German mathematician Felix Hausdorff (1868–1942).

Hausdorff space (plural Hausdorff spaces)

1. (topology) A topological space in which for any two distinct points x and y, there is a pair of disjoint open sets U and V such that ${\displaystyle x\in U}$ and ${\displaystyle y\in V}$ .
   • 2005, N. L. Carothers, A Short Course on Banach Space Theory, Cambridge University Press, page 167,

     More generally, each compact subset of a Hausdorff space is closed. […] Metric spaces and compact Hausdorff spaces enjoy an even stronger separation property; in either case, disjoint closed sets can always be separated by disjoint open sets.

   • 2008, Thiruvaiyaru V. Panchapagesan, The Bartle-Dunford-Schwartz Integral, Springer (Birkhäuser), page ix,

     In 1953, Grothendieck [G] characterized locally convex Hausdorff spaces which have the Dunford-Pettis property and used this property to characterize weakly compact operators u : C(K) → F, where K is a compact Hausdorff space and F is a locally compact Hausdorff space (briefly, lcHs) which is complete.

   • 2012, Neil Hindman, Dona Strauss,Algebra in the Stone-Cech Compactification: Theory and Applications, Walter de Gruyter, 2nd Edition, page 83,

     Our construction of βD is a special case of more general constructions in which compact Hausdorff spaces are obtained using sets which are maximal subject to having certain algebraic properties.

• (topological space in which distinct points are contained in distinct open sets): T₂ space

• sober space
  • topological space

• metric space
• normal space
• regular space