- Sober space
In
mathematics , particularly intopology , a sober space is a particular kind oftopological space .Specifically, a space "X" is "sober" if every irreducible
closed subset of "X" is the closure of exactly one singleton of "X": that is, has a uniquegeneric point .An "irreducible" closed subset of "X" is anonempty closed subset of "X" which is not the union of two proper closed subsets of itself.Any Hausdorff (T2) space is sober (the only irreducible subsets being points), and allsober spaces are Kolmogorov (T0). Sobriety isnot comparable to the T1 condition.
The
prime spectrum Spec("R") of acommutative ring "R" with theZariski topology is acompact sober space. In fact, every compact sober space is homeomorphic to Spec("R") for some commutative ring "R".Fact|date=July 2008Sobriety of "X" is precisely a condition that forces the lattice of open subsets of "X" to determine "X"
up to homeomorphism , which is relevant topointless topology .Sobriety makes the
specialization preorder adirected complete partial order .ee also
*
Stone duality , on the duality between topological spaces which are sober and frames (i.e.complete Heyting algebra s) which are spatial.External links
* [http://www.mathematik.tu-darmstadt.de:8080/Math-Net/Lehrveranstaltungen/Lehrmaterial/SS2003/Topology/separation.pdf Discussion of weak separation axioms] (PDF file)
References
* Steven Vickers, "Topology via logic",
Cambridge University Press , 1989, ISBN 0-521-36062-5. Page 66.
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