Sober space
In mathematics, particularly in topology, a sober space is a particular kind of topological space. Specifically, a space X is sober if every irreducible closed subset[1] of X is the closure of exactly one singleton of X: that is, has a unique generic point.
Properties
Any Hausdorff (T2) space is sober (the only irreducible subsets being points), and all sober spaces are Kolmogorov (T0). Sobriety is not comparable to the T1 condition.
The prime spectrum Spec(R) of a commutative ring R with the Zariski topology is a compact sober space. In fact, every compact sober space is homeomorphic to Spec(R) for some commutative ring R. This is a theorem of Melvin Hochster.
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.
Notes
- ^ An irreducible closed subset of X is a nonempty closed subset of X which is not the union of two proper closed subsets of itself.
See also
- Stone duality, on the duality between topological spaces which are sober and frames (i.e. complete Heyting algebras) which are spatial.
External links
- 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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