Sober space

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.