Sober space

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In general topology and logic, a sober space is 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 spaces 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.