High Quality Content by WIKIPEDIA articles! In mathematics, particularly in topology, a sober space is a particular kind of topological space. Specifically, a space X is sober if every irreducible closed subset of X is the closure of exactly one singleton of X: that is, has a unique generic point.Any Hausdorff (T2) space is sober (the only irreducible subsets being points), and all sober spaces are Kolmogorov (T0). Sobriety is not comparable to the T1 condition. More precisely, T2 is equivalent to T1 and sober. Indeed, in any topological space the intersection of all closed neighborhoods of a point is always an irreducible (non-empty) closed subset; if the space is sober, this irreducible closed set is the closure of a point, which is reduced to the point itself if the space is also T1. The latter is indeed a characterization of T2 spaces: a space is T2 if and only if the intersection of all closed neighborhoods of any point is reduced to the point itself.

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