Extremally disconnected space

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search

In mathematics, an extremally disconnected space is a topological space in which the closure of every open set is open. (The term "extremally disconnected" is correct, even though the word "extremally" does not appear in most dictionaries.[1] The term extremely disconnected is sometimes used, but it is incorrect.)

An extremally disconnected space that is also compact and Hausdorff is sometimes called a Stonean space. This is different from a Stone space, which is usually a totally disconnected compact Hausdorff space. A theorem due to Andrew Gleason says that the projective objects of the category of compact Hausdorff spaces are exactly the extremally disconnected compact Hausdorff spaces. In the duality between Stone spaces and Boolean algebras, the Stonean spaces correspond to the complete Boolean algebras.

An extremally disconnected first-countable collectionwise Hausdorff space must be discrete. In particular, for metric spaces, the property of being extremally disconnected (the closure of every open set is open) is equivalent to the property of being discrete (every set is open).

Examples[edit]

Applications[edit]

Hartig (1983) proves the Riesz–Markov–Kakutani representation theorem by reducing it to the case of extremally disconnected spaces, in which case the representation theorem can be proved by elementary means.

References[edit]

  • A. V. Arkhangelskii (2001) [1994], "Extremally-disconnected space", in Hazewinkel, Michiel (ed.), Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
  • Johnstone, Peter T (1982). Stone spaces. Cambridge University Press. ISBN 0-521-23893-5.
  • Hartig, Donald G. (1983), "The Riesz representation theorem revisited", American Mathematical Monthly, 90 (4): 277–280, doi:10.2307/2975760