sober space

{{Short description|Topological space whose topology is fully captured by its lattice of open sets}}

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.{{cite book |last1=Mac Lane |first1=Saunders |title=Sheaves in geometry and logic: a first introduction to topos theory |date=1992 |publisher=Springer-Verlag |location=New York |isbn=978-0-387-97710-2 |pages=472–482}}{{cite journal |last1=Sünderhauf |first1=Philipp |title=Sobriety in Terms of Nets |journal=Applied Categorical Structures |date=1 December 2000 |volume=8 |issue=4 |pages=649–653 |doi=10.1023/A:1008673321209}} 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 [[Complete Heyting algebra|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 $x$ .

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.{{cite book | title=Encyclopedia of general topology | url=https://archive.org/details/encyclopediagene00hart_882 | url-access=limited | first1=Klaas Pieter | last1=Hart | first2=Jun-iti | last2=Nagata | first3=Jerry E. | last3=Vaughan | publisher=Elsevier | year=2004 | isbn=978-0-444-50355-8 | pages=[https://archive.org/details/encyclopediagene00hart_882/page/n165 155]–156 }}

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.{{Cite web|url=https://math.stackexchange.com/questions/4198805|title=General topology - Finite $T_0$ 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.{{Citation

| last = Hochster

| first = Melvin

| title =Prime ideal structure in commutative rings

| journal = Trans. Amer. Math. Soc.

| volume = 142

| year = 1969

| pages = 43–60

| doi=10.1090/s0002-9947-1969-0251026-x

| doi-access= free

}}

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.