In mathematics, a sober space is a topological space X such that every (nonempty) irreducible closed subset of X is the closure of exactly one point of X: that is, every nonempty irreducible closed subset has a unique generic point.
Sober spaces have a variety of cryptomorphic definitions, which are documented in this section [1] [2].
In each case below, replacing "unique" with "at most one" gives an equivalent formulation of the T0 axiom.
Replacing it with "at least one" is equivalent to the property that the T0 quotient of the space is sober, which is sometimes referred to as having "enough points" in the literature.
A space is sober if every nonempty irreducible closed subset is the closure of a unique point.
A topological space X is sober if every map that preserves all joins and all finite meets from its partially ordered set of open subsets to
is the inverse image of a unique continuous function from the one-point space to X.
[2] In particular, a space is T1 and sober precisely if every self-convergent net is constant.
A space X is sober if every functor from the category of sheaves Sh(X) to Set that preserves all finite limits and all small colimits must be the stalk functor of a unique point x.
Sobriety makes the specialization preorder a directed complete partial order.
Every continuous directed complete poset equipped with the Scott topology is sober.
[4] The prime spectrum Spec(R) of a commutative ring R with the Zariski topology is a compact sober space.
[3] In fact, every spectral space (i.e. a compact sober space for which the collection of compact open subsets is closed under finite intersections and forms a base for the topology) is homeomorphic to Spec(R) for some commutative ring R. This is a theorem of Melvin Hochster.
The subset of Spec(R) consisting only of the maximal ideals, where R is a commutative ring, is not sober in general.