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. 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, 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 is a compact sober space (with the Zariski topology). In fact, every compact sober space is homeomorphic to Spec(R) for some commutative ring R.
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] References
- Discussion of weak separation axioms (PDF file)
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