Sober space

In mathematics, a sober space is a topological space X such that every 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.

Definitions

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 T₀ axiom. Replacing it with "at least one" is equivalent to the property that the T₀ quotient of the space is sober, which is sometimes referred to as having "enough points" in the literature.

With irreducible closed sets

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.

In terms of morphisms of frames and locales">Complete Heyting algebra">frames and locales

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.

Using completely prime filters

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.

In terms of nets

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 T₁ and sober precisely if every self-convergent net is constant.

As a property of sheaves on the space

A space X is sober if every functor from the category of sheaves Sh to Set that preserves all finite limits and all small colimits must be the stalk functor of a unique point x.

Properties and examples

Any Hausdorff space is sober, and all sober spaces are Kolmogorov, and both implications are strict.
Sobriety is not comparable to the T₁ condition:

an example of a T₁ space that is not sober is an infinite set with the cofinite topology, the whole space being an irreducible closed subset with no generic point;
an example of a sober space that is not T₁ is the Sierpinski space.

Moreover, T₂ is strictly stronger than T₁ and sober, i.e., while every T₂ space is at once T₁ and sober, there exist spaces that are simultaneously T₁ and sober, but not T₂. 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 T₀ spaces are sober.
The prime spectrum Spec of a commutative ring R with the Zariski topology is a compact sober space. In fact, every spectral space is homeomorphic to Spec 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 consisting only of the maximal ideals, where R is a commutative ring, is not sober in general.