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Difference between revisions of "Sober space"

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{{MSC|54Dxx}}
 
{{MSC|54Dxx}}
  
A  [[topological space]] in which every [[Irreducible topological space|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.   
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A  [[topological space]] in which every irreducible [[closed set]] has a unique [[generic point]].  Here a closed set is ''[[Irreducible topological space|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 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.
 
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.   
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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.  Every subset of a sober [[TD space]] is sober.
  
 
==References==
 
==References==
*  Peter T. Johnstone;        ''Sketches of an elephant'',    ser. Oxford Logic Guides  (2002) Oxford University Press. ISBN 0198534256 pp. 491-492
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*  Peter T. Johnstone;        ''Sketches of an elephant'',    ser. Oxford Logic Guides  (2002) Oxford University Press. {{ISBN|0198534256}} pp. 491-492 {{ZBL|1071.18001}}
*  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   
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*  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'' Cambridge Tracts in Theoretical Computer Science '''5''' Cambridge University Press (1989) ISBN 0-521-36062-5 {{ZBL|0668.54001}}. p.66
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*  Steven Vickers ''Topology via Logic'' Cambridge Tracts in Theoretical Computer Science '''5''' Cambridge University Press (1989) {{ISBN|0-521-36062-5}} {{ZBL|0668.54001}}. p.66

Latest revision as of 16:50, 4 November 2023

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. Every subset of a sober TD space is sober.

References

  • Peter T. Johnstone; Sketches of an elephant, ser. Oxford Logic Guides (2002) Oxford University Press. ISBN 0198534256 pp. 491-492 Zbl 1071.18001
  • 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 Cambridge Tracts in Theoretical Computer Science 5 Cambridge University Press (1989) ISBN 0-521-36062-5 Zbl 0668.54001. p.66
How to Cite This Entry:
Sober space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sober_space&oldid=37225