irreducible closed subspace

Given a topological space XX, a closed subspace FF of XX is irreducible if there are exactly two ways to express FF as a union of two closed subspaces: F=FF = \empty \cup F and F=FF = F \cup \empty. In other words, FF must be inhabited (so that these two ways are distinct) but it must be impossible to express FF as a union of two inhabited closed subspaces.

Note that the closure of any point of XX is an irreducible closed subspace. XX is sober if and only if every irreducible closed subspace is the closure of a unique point of XX. In general, the irreducible closed subspaces of XX correspond to the points of the topological locale Ω(X)\Omega(X).

The theory of irreducible closed subspaces is not useful in constructive mathematics; instead, one should use the completely prime filters on the frame of open subspaces of XX (which are the points of Ω(X)\Omega(X) by definition).

Created on July 1, 2010 17:25:43 by Toby Bartels (