Sober space
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In mathematics, particularly in topology, a topological 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 defined to be a nonempty closed subset of X which is not the union of two proper closed subsets of itself.
Any Hausdorff (<math>T_2<math>) space is sober, and all sober spaces are Kolmogorov (<math>T_0<math>). Sobriety is not comparable to T1.
Sobriety of X is precisely the condition that forces the lattice of open subsets of X to determine X up to homeomorphism.
Sobriety makes the specialization preorder a DCPO.
See also pointless topology.
External link
- Discussion of weak separation axioms (http://www.mathematik.tu-darmstadt.de/Math-Net/Lehrveranstaltungen/Lehrmaterial/SS2003/Topology/separation.pdf) (PDF file)