, particularly in topology
, a sober space
is a particular kind of topological space
Specifically, a space X is sober if every irreducible closed subset of X is the closure of exactly one singleton of X: that is, has a unique generic point.
An irreducible closed subset of X is a nonempty closed subset of X which is not the union of two proper closed subsets of itself.
Any Hausdorff (T2) space is sober (the only irreducible subsets being points), and all
sober spaces are Kolmogorov (T0). Sobriety is
not comparable to the T1 condition.
The prime spectrum Spec(R) of a commutative ring R with the Zariski topology is a compact sober space. In fact, every compact sober space is homeomorphic to Spec(R) for some commutative ring R.
Sobriety of X is precisely a condition that forces the lattice of open subsets of X to determine X up to homeomorphism, which is relevant to pointless topology.
Sobriety makes the specialization preorder a directed complete partial order.