Sober space: Difference between revisions

Revision as of 19:08, 17 April 2004

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In [[mathematics]], particularly in [[topology]], a [[topological space]] ''X'' is ''sober'' if for all [[closed]] [[subset | subsets]] ''C'' of ''X'' strictly containing no smaller [[nonempty]] closed [[set]], [[there exists]] a [[Point_(topology) | point]] ''x'' in ''X'' [[such that]] ''C'' is the [[Topological_closure | closure]] of the [[Singleton_mathematicsSingleton_(mathematics) | singleton]] {''x''}.

Any [[T2_space | Hausdorff]] (<math>T_2</math>) space is sober, and all sober spaces are [[T0_space | Kolmogorov]] (<math>T_0</math>). Sobriety is not comparable to [[T1_space | <math>T_1</math>]].