View source for Sober space

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{{MSC|54Dxx}}

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.

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==
*  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