# Hausdorff space

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## English[edit]

### Etymology[edit]

After German mathematician Felix Hausdorff (1868–1942).

### Noun[edit]

**Hausdorff space** (*plural* **Hausdorff spaces**)

- (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 and .**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.

- More generally, each compact subset of a
**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.

- In 1953, Grothendieck [G] characterized locally convex
**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.

- Our construction of

#### Synonyms[edit]

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