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 (T₂) space is sober, and all sober spaces are Kolmogorov (T₀). 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 (http://www.mathematik.tu-darmstadt.de/Math-Net/Lehrveranstaltungen/Lehrmaterial/SS2003/Topology/separation.pdf) (PDF file)