Given a topological space , a closed subspace of is irreducible if there are exactly two ways to express as a union of two closed subspaces: and . In other words, must be inhabited (so that these two ways are distinct) but it must be impossible to express as a union of two inhabited closed subspaces.
Note that the closure of any point of is an irreducible closed subspace. is sober if and only if every irreducible closed subspace is the closure of a unique point of . In general, the irreducible closed subspaces of correspond to the points of the topological locale .
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 (which are the points of by definition).