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 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 A topological space X is sober if every map from its partially ordered set of open subsets to {0, 1} 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 O i {\displaystyle O_{i}} of open sets such that ⋃ i O i ∈ F {\displaystyle \bigcup _{i}O_{i}\in F} , we have that O i ∈ F {\displaystyle O_{i}\in F} 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 x ∙ {\displaystyle x_{\bullet }} is self-convergent if it converges to every point x i {\displaystyle x_{i}} in x ∙ {\displaystyle x_{\bullet }} , or equivalently if its eventuality filter is completely prime. A net x ∙ {\displaystyle x_{\bullet }} that converges to x {\displaystyle x} converges strongly if it can only converge to points in the closure of x {\displaystyle x} . A space is sober if every self-convergent net x ∙ {\displaystyle x_{\bullet }} converges strongly to a unique point x {\displaystyle x} . 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(X) 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 (T2) space is sober (the only irreducible subsets being singletons), and all sober spaces are Kolmogorov (T0), 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(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.
See also Stone duality, on the duality between topological spaces that are sober and frames (i.e. complete Heyting algebras) that are spatial.
References
Further reading Pedicchio, Maria Cristina; Tholen, Walter, eds. (2004). Categorical foundations. Special topics in order, topology, algebra, and sheaf theory. Encyclopedia of Mathematics and Its Applications. Vol. 97. Cambridge: Cambridge University Press. ISBN 0-521-83414-7. Zbl 1034.18001. Vickers, Steven (1989). Topology via logic. Cambridge Tracts in Theoretical Computer Science. Vol. 5. Cambridge: Cambridge University Press. p. 66. ISBN 0-521-36062-5. Zbl 0668.54001.
