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 irreducible closed subset has a unique generic point.

Sober spaces have a variety of cryptomorphic definitions, which are documented in this section. All except the definition in terms of nets are described in.^{[1]} In each case below, replacing "unique" with "at most one" gives an equivalent formulation of the T_{0} axiom. Replacing it with "at least one" is equivalent to the property that the T_{0} quotient of the space is sober, which is sometimes referred to as having "enough points" in the literature.

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*.

This may be viewed as a correspondence between the notion of a point in a locale and a point in a topological space, which is the motivating definition.

A filter *F* of open sets is said to be *completely prime* if for any family of open sets such that , we have that for some *i*. A space X is sober if it each completely prime filter is the neighbourhood filter of a unique point in X.

A net is *self-convergent* if it converges to every point in , or equivalently if its eventuality filter is completely prime. A net that converges to *converges strongly* if it can only converge to points in the closure of . A space is sober if every self-convergent net converges strongly to a unique point .^{[2]}

In particular, a space is T1 and sober precisely if every self-convergent net is constant.

A closed set is irreducible if it cannot be written as the union of two proper closed subsets. A space is sober if every irreducible closed subset is the closure of a unique point.

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*.

Any Hausdorff (T_{2}) space is sober (the only irreducible subsets being points), and all sober spaces are Kolmogorov (T_{0}), and both implications are strict.^{[3]}

Sobriety is not comparable to the T_{1} condition:

- an example of a T
_{1}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; - an example of a sober space which is not T
_{1}is the Sierpinski space.

Moreover T_{2} is stronger than T_{1} *and* sober, i.e., while every T_{2} space is at once T_{1} and sober, there exist spaces that are simultaneously T_{1} and sober, but not T_{2}. 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.

Sobriety makes the specialization preorder a directed complete partial order.

Every continuous directed complete poset equipped with the Scott topology is sober.

Finite T_{0} spaces are 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.^{[5]}
More generally, the underlying topological space of any scheme is a sober space.

The subset of Spec(*R*) consisting only of the maximal ideals, where *R* is a commutative ring, is not sober in general.