Sober space

Summary

A sober space is a topological space in which every irreducible closed set is the closure of exactly one point. That point is called the generic point of the set. Every sober space is $T_0$, and every Hausdorff space is sober. Spectra of commutative rings with the Zariski topology are sober. A sober space can be recovered, up to homeomorphism, from its lattice of open sets.