Sober space: Difference between revisions
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In [[general topology]] and [[logic]], a '''sober space''' is 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. | In [[general topology]] and [[logic]], a '''sober space''' is 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. | ||
Revision as of 12:09, 7 February 2009
In general topology and logic, a sober space is 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 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.
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.
References
- Peter T. Johnstone (2002). Sketches of an elephant. Oxford University Press, 491-492. ISBN 0198534256.
- Maria Cristina Pedicchio; Walter Tholen (2004). Categorical Foundations: Special Topics in Order, Topology, Algebra, and Sheaf Theory. Cambridge University Press, 54-55. ISBN 0-521-83414-7.
- Steven Vickers (1989). Topology via Logic. Cambridge University Press, 66. ISBN 0-521-36062-5.