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Sober space
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 (T2) space is sober, and all sober spaces are Kolmogorov (T0). 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 (PDF file)
Last updated: 08-02-2005 05:06:24
10-26-2009 08:16:03
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The contents of this article is licensed from www.wikipedia.org under the GNU Free Documentation License. Click here to see the transparent copy and copyright details