Sober space

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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 of X is the closure of exactly one singleton of X: that is, has a unique generic point. 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.

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.^{[citation needed]}

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.

[edit] See also

• Stone duality, on the duality between topological spaces which are sober and frames (i.e. complete Heyting algebras) which are spatial.

[edit] External links

• Discussion of weak separation axioms (PDF file)

[edit] References

• Steven Vickers, Topology via logic, Cambridge University Press, 1989, ISBN 0-521-36062-5. Page 66.