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. | ||
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. | 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. | ||
A sober space is characterised by its [[lattice (order)|lattice]] of [[open set]]s. An open set in a sober space is again a sober space, as is a closed set. | |||
==References== | ==References== | ||
* {{cite book | author=Peter T. Johnstone | title=Sketches of an elephant | series=Oxford Logic Guides | publisher=[[Oxford University Press]] | year=2002 | isbn=0198534256 | pages=491-492 }} | |||
* {{cite book | author=Maria Cristina Pedicchio | coauthors=Walter Tholen | title=Categorical Foundations: Special Topics in Order, Topology, Algebra, and Sheaf Theory | publisher=[[Cambridge University Press]] | year=2004 | isbn=0-521-83414-7 | pages=54-55 }} | |||
* {{cite book | author=Steven Vickers | title=Topology via Logic | publisher=[[Cambridge University Press]] | year=1989 | isbn=0-521-36062-5 | pages=66 }} | * {{cite book | author=Steven Vickers | title=Topology via Logic | publisher=[[Cambridge University Press]] | year=1989 | isbn=0-521-36062-5 | pages=66 }} | ||
Revision as of 15:25, 4 January 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.