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.
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 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.
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
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 <math>O_i</math> of open sets such that <math>\bigcup_i O_i \in F</math>, we have that <math>O_i \in F</math> 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 <math>x_{\bullet}</math> is self-convergent if it converges to every point <math>x_i</math> in <math>x_{\bullet}</math>, or equivalently if its eventuality filter is completely prime. A net <math>x_{\bullet}</math> that converges to <math>x</math> converges strongly if it can only converge to points in the closure of <math>x</math>. A space is sober if every self-convergent net <math>x_{\bullet}</math> converges strongly to a unique point <math>x</math>.
Sobriety is not comparable to the T<sub>1</sub> condition:
- an example of a T<sub>1</sub> 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 T<sub>1</sub> is the Sierpinski space.
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.
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.
See also
- Stone duality, on the duality between topological spaces that are sober and frames (i.e. complete Heyting algebras) that are spatial.