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. 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.
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. 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.
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- 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.