# 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, this closed subset has a unique generic point.

## Properties and examples

Any Hausdorff (T_{2}) space is sober (the only irreducible subsets being points), and all sober spaces are Kolmogorov (T_{0}), and both implications are strict.[1]
Sobriety is not comparable to the T_{1} condition: an example of a T_{1} 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 T_{2} is stronger than T_{1} *and* sober, i.e., while every T_{2} space is at once T_{1} and sober, there exist spaces that are simultaneously T_{1} and sober, but not T_{2}. 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.

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

- Hart, Klaas Pieter; Nagata, Jun-iti; Vaughan, Jerry E. (2004).
*Encyclopedia of general topology*. Elsevier. pp. 155–156. ISBN 978-0-444-50355-8. - 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.