extremally disconnected

A topological spaceMathworldPlanetmath X is said to be extremally disconnected if every open set in X has an open closurePlanetmathPlanetmath.

It can be shown that X is extremally disconnected iff any two disjoint open sets in X have disjoint closures. Every extremally disconnected space is totally disconnected.


Some authors like [1] and [2] use the above definition as is, while others (e.g. [3, 4]) require that an extremally disconnected space should (in addition to the above condition) also be a Hausdorff space.


  • 1 S. Willard, General Topology, Addison-Wesley, Publishing Company, 1970.
  • 2 J. L. Kelley, General Topology, D. van Nostrand Company, Inc., 1955.
  • 3 L. A. Steen, J. A. Seebach, Jr., Counterexamples in topology, Holt, Rinehart and Winston, Inc., 1970.
  • 4 N. Bourbaki, General Topology, Part 1, Addison-Wesley Publishing Company, 1966.
Title extremally disconnected
Canonical name ExtremallyDisconnected
Date of creation 2013-03-22 12:42:00
Last modified on 2013-03-22 12:42:00
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 8
Author PrimeFan (13766)
Entry type Definition
Classification msc 54G05
Synonym extremely disconnected
Related topic ConnectedSpace