Sober space

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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 irreducible closed subset has a unique generic point.


Properties and examples

All Hausdorff (T2) spaces are sober (the only irreducible subsets being points), and all sober spaces are Kolmogorov (T0) spaces. Both implications are strict. [1]

T1 spaces need not be sober. 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. Conversely, the Sierpinski space is an example of a sober spaces which is not a T1 space.

All T2 spaces are necessarily both T1 and sober, but 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

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  1. 1 2 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

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