Sober space

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 ^[1] ^[2]. 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 that preserves all joins and all finite meets from its partially ordered set of open subsets to $\{0,1\}$ 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 $O_{i}$ of open sets such that $\bigcup _{i}O_{i}\in F$ , we have that $O_{i}\in F$ 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 $x_{\bullet }$ is self-convergent if it converges to every point $x_{i}$ in $x_{\bullet }$ , or equivalently if its eventuality filter is completely prime. A net $x_{\bullet }$ that converges to $x$ converges strongly if it can only converge to points in the closure of $x$ . A space is sober if every self-convergent net $x_{\bullet }$ converges strongly to a unique point $x$ .^[2]

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 points), and all sober spaces are Kolmogorov (T₀), and both implications are strict.^[3]

Sobriety is not comparable to the T₁ condition:

an example of a T₁ space which 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 which is not T₁ is the Sierpinski space.

Moreover T₂ is 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.^[4]

The prime spectrum Spec(R) of a commutative ring R with the Zariski topology is a compact sober space.^[3] 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.^[5] 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.

References

^ Mac Lane, Saunders (1992). Sheaves in geometry and logic: a first introduction to topos theory. New York: Springer-Verlag. pp. 472–482. ISBN 978-0-387-97710-2.
^ ^a ^b Sünderhauf, Philipp (1 December 2000). "Sobriety in Terms of Nets". Applied Categorical Structures. 8 (4): 649–653. doi:10.1023/A:1008673321209.
^ ^a ^b Hart, Klaas Pieter; Nagata, Jun-iti; Vaughan, Jerry E. (2004). Encyclopedia of general topology. Elsevier. pp. 155–156. ISBN 978-0-444-50355-8.
^ "General topology - Finite $T_0$ spaces are sober".
^ Hochster, Melvin (1969), "Prime ideal structure in commutative rings", Trans. Amer. Math. Soc., 142: 43–60, doi:10.1090/s0002-9947-1969-0251026-x