Sober space

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