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 irreducible closed subset has a unique generic point.

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.[1] 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. 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.

The prime spectrum Spec(R) of a commutative ring R with the Zariski topology is a compact sober space.[1] 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.[2] 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 which are sober and frames (i.e. complete Heyting algebras) which are spatial.

References

1. Hart, Klaas Pieter; Nagata, Jun-iti; Vaughan, Jerry E. (2004). Encyclopedia of general topology. Elsevier. pp. 155–156. ISBN 978-0-444-50355-8.
2. Hochster, Melvin (1969), "Prime ideal structure in commutative rings", Trans. Amer. Math. Soc., 142: 43–60, doi:10.1090/s0002-9947-1969-0251026-x

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. 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. 5. Cambridge: Cambridge University Press. p. 66. ISBN 0-521-36062-5. Zbl 0668.54001.