Hausdorff space

Noun

Hausdorff space (plural Hausdorff spaces)

1. 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 x∈U and y∈V.
   • 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.2005, N. L. Carothers, A Short Course on Banach Space Theory, Cambridge University Press, page 167
   • 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...2008, Thiruvaiyaru V. Panchapagesan, The Bartle-Dunford-Schwartz Integral, Springer (Birkhäuser), page ix

   Synonyms:

   T₂ space