In mathematics, a sober space is a topological space X such that every (nonempty) irreducible closed subset of X is the closure of exactly one point of X: that is, every nonempty irreducible closed subset has a unique generic point.
Sober spaces have a variety of cryptomorphic definitions, which are documented in this section. In each case below, replacing "unique" with "at most one" gives an equivalent formulation of the T<sub>0</sub> axiom. Replacing it with "at least one" is equivalent to the property that the T<sub>0</sub> quotient of the space is sober, which is sometimes referred to as having "enough points" in the literature.
A closed set is irreducible if it cannot be written as the union of two proper closed subsets. A space is sober if every nonempty irreducible closed subset is the closure of a unique point.
A topological space X is sober if every map from its partially ordered set of open subsets to that preserves all joins and all finite meets is the inverse image of a unique continuous function from the one-point space to X.
This may be viewed as a correspondence between the notion of a point in a locale and a point in a topological space, which is the motivating definition.
A filter F of open sets is said to be completely prime if for any family of open sets such that , we have that for some i. A space X is sober if each completely prime filter is the neighbourhood filter of a unique point in X.
A net is self-convergent if it converges to every point in , or equivalently if its eventuality filter is completely prime. A net that converges to converges strongly if it can only converge to points in the closure of . A space is sober if every self-convergent net converges strongly to a unique point .
In particular, a space is T<sub>1</sub> and sober precisely if every self-convergent net is constant.
A space X is sober if every functor from the category of sheaves Sh(X) to Set that preserves all finite limits and all small colimits must be the stalk functor of a unique point x.
Any Hausdorff (T<sub>2</sub>) space is sober (the only irreducible subsets being singletons), and all sober spaces are Kolmogorov (T<sub>0</sub>), and both implications are strict.
Sobriety is not comparable to the T<sub>1</sub> condition:
Moreover, T<sub>2</sub> is strictly stronger than T<sub>1</sub> and sober, i.e., while every T<sub>2</sub> space is at once T<sub>1</sub> and sober, there exist spaces that are simultaneously T<sub>1</sub> and sober, but not T<sub>2</sub>. One such example is the following: let X be the set of real numbers, with a new point p adjoined; the open sets being all real open sets, and all cofinite sets containing p.
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.
Every continuous directed complete poset equipped with the Scott topology is sober.
Finite T<sub>0</sub> spaces are sober.
The prime spectrum Spec(R) of a commutative ring R with the Zariski topology is a compact sober space. In fact, every spectral space (i.e. a compact sober space for which the collection of compact open subsets is closed under finite intersections and forms a base for the topology) is homeomorphic to Spec(R) for some commutative ring R. This is a theorem of Melvin Hochster. More generally, the underlying topological space of any scheme is a sober space.
The subset of Spec(R) consisting only of the maximal ideals, where R is a commutative ring, is not sober in general.