Sober space

In mathematics, a sober space is a topological space such that every irreducible closed subset of X is the closure of exactly one point of X: that is, this closed subset has a unique generic point.

Properties and examples

Any Hausdorff (T₂) space is sober (the only irreducible subsets being points), and all sober spaces are Kolmogorov (T₀), and both implications are strict. Sobriety is not comparable to the T₁ condition: an example of a T₁ space which is not sober is an infinite set with the cofinite topology, the whole space being an irreducible closed subset with no generic point. Moreover T₂ is stronger than T₁and sober, i.e., while every T₂ space is at once T₁ and sober, there exist spaces that are simultaneously T₁ and sober, but not T₂. One such example is the following: let X be the set of real numbers, with a new point p adjoined; the open sets being all real open sets, and all cofinite sets containing p.

Sobriety of X is precisely a condition that forces the lattice of open subsets of X to determine X up to homeomorphism, which is relevant to pointless topology.