Sober space

In mathematics, particularly in topology, a topological space X is sober if for all closed subsets C of X strictly containing no smaller nonempty closed set, there exists a point x in X such that C is the closure of the singleton {x}.

Any Hausdorff (${\displaystyle T_{2}}$) space is sober, and all sober spaces are Kolmogorov (${\displaystyle T_{0}}$). Sobriety is not comparable to ${\displaystyle T_{1}}$.

Sobriety of X is precisely the condition that forces the ring ${\displaystyle C^{0}(X,\mathbb {R} )}$ of continuous real-valued functions on X to determine X up to homeomorphism.

Sobriety makes the specialization preorder a DCPO.

Discussion of weak separation axioms