Sober space

In mathematics, particularly in topology, a topological space X is sober if every irreducible closed subset of X is the closure of exactly one singleton of X. An irreducible closed subset of X is defined to be a nonempty closed subset of X which is not the union of two proper closed subsets of itself.

Any Hausdorff (<math>T_2<math>) space is sober, and all sober spaces are Kolmogorov (<math>T_0<math>). Sobriety is not comparable to T₁.

Sobriety of X is precisely the condition that forces the lattice of open subsets of X to determine X up to homeomorphism.

Sobriety makes the specialization preorder a DCPO.

See also pointless topology.

External link

• Discussion of weak separation axioms (PDF file)