In mathematics, particularly in topology, a topological space X is sober if every irreducible closed subset of X is the closure of exactly one singleton of X.
An irreducible closed subset of X is defined to be a nonempty closed subset of X which is not the union of two proper closed subsets of itself.
Any Hausdorff (T2) space is sober, and all
sober spaces are Kolmogorov (T0). Sobriety is
not comparable to T1.
Sobriety of X is precisely the condition that forces the lattice of open subsets of X to determine X up to homeomorphism.
Sobriety makes the specialization preorder
a DCPO.
See also pointless topology.
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