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. More precisely, T₂ is equivalent to T₁ and sober. Indeed, in any topological space the intersection of all closed neighborhoods of a point is always an irreducible (non-empty) closed subset; if the space is sober, this irreducible closed set is the closure of a point, which is reduced to the point itself if the space is also T₁. The latter is indeed a characterization of T₂ spaces: a space is if T₂ if and only if the intersection of all closed neighborhoods of any point is reduced to the point itself.

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

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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