Difference between revisions of "Sober space"

Revision as of 20:52, 5 December 2014

2020 Mathematics Subject Classification: Primary: 54Dxx [MSN][ZBL]

A topological space in which every irreducible closed set has a unique generic point. Here a closed set is irreducible if it is not the union of two non-empty proper closed subsets of itself.

Any Hausdorff space is sober, since the only irreducible subsets are singletons. Any sober space is a T0 space. However, sobriety is not equivalent to the T1 space condition: an infinite set with the cofinite topology is T1 but not sober whereas a Sierpinski space is sober but not T1.

A sober space is characterised by its lattice of open sets. An open set in a sober space is again a sober space, as is a closed set.

Peter T. Johnstone; Sketches of an elephant, ser. Oxford Logic Guides (2002) Oxford University Press. ISBN 0198534256 pp. 491-492
Maria Cristina Pedicchio; Walter Tholen; Categorical Foundations: Special Topics in Order, Topology, Algebra, and Sheaf Theory, (2004) Cambridge University Press ISBN 0-521-83414-7. pp. 54-55
Steven Vickers; Topology via Logic, (1989) Cambridge University Press ISBN 0-521-36062-5. p.66

How to Cite This Entry:
Sober space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sober_space&oldid=35377

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 {{MSC|54Dxx}}
-A  [[topological space]] in which every [[irreducible set|irreducible]]  [[closed set]] has a unique [[generic point]].  Here a closed set is  ''irreducible'' if it is not the union of two non-empty proper closed  subsets of itself.
+A  [[topological space]] in which every [[irreducible set|irreducible]]  [[closed set]] has a unique [[generic point]].  Here a closed set is  ''irreducible'' if it is not the union of two non-empty proper closed subsets of itself.
-Any [[Hausdorff space]] is sober,  since the only irreducible subsets are [[singleton]]s.  Any sober spaces  is a [[T0 space]].  However, sobriety is not equivalent to the [[T1  space]] condition: an infinite set with the [[cofinite topology]] is T1  but not sober whereas a [[Sierpinski space]] is sober but not T1.
+Any [[Hausdorff space]] is sober,  since the only irreducible subsets are [[singleton]]s.  Any sober space  is a [[T0 space]].  However, sobriety is not equivalent to the [[T1 space]] condition: an infinite set with the [[cofinite topology]] is T1 but not sober whereas a [[Sierpinski space]] is sober but not T1.
 A  sober space is characterised by its [[lattice]] of  [[open set]]s.   An open set in a sober space is again a sober space, as  is a closed set.