sober space

Let X be a topological spaceMathworldPlanetmath. A subset A of X is said to be irreducible if whenever ABC with B,C closed, we have AB or AC. Any singleton and its closurePlanetmathPlanetmath are irreducible. More generally, the closure of an irreducible set is irreducible.

A topological space X is called a sober space if every irreducible closed subset is the closure of some unique point in X.


  • For any sober space, the closure of a point determines the point. In other words, cl(x)=cl(y) implies x=y.

  • A space is sober iff the closure of every irreducible set is the closure of a unique point.

  • Any sober space is T0.

  • Any Hausdorff space is sober.

  • A closed subspace of a sober space is sober.

  • Any product of sober spaces is sober.

Title sober space
Canonical name SoberSpace
Date of creation 2013-03-22 16:43:44
Last modified on 2013-03-22 16:43:44
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 6
Author CWoo (3771)
Entry type Definition
Classification msc 54E99
Defines irreducible set