# Sober space

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.

## Definitions

Sober spaces have a variety of cryptomorphic definitions, which are documented in this section. 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.

### With irreducible closed sets

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

### In terms of morphisms of frames and locales

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 $\{0,1\}$ 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.

### Using completely prime filters

A filter F of open sets is said to be completely prime if for any family $O_{i}$ of open sets such that $\bigcup _{i}O_{i}\in F$, we have that $O_{i}\in F$ for some i. A space X is sober if each completely prime filter is the neighbourhood filter of a unique point in X.

### In terms of nets

A net $x_{\bullet }$ is self-convergent if it converges to every point $x_{i}$ in $x_{\bullet }$, or equivalently if its eventuality filter is completely prime. A net $x_{\bullet }$ that converges to $x$ converges strongly if it can only converge to points in the closure of $x$. A space is sober if every self-convergent net $x_{\bullet }$ converges strongly to a unique point $x$.
In particular, a space is T1 and sober precisely if every self-convergent net is constant.

### As a property of sheaves on the space

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.

## Properties and examples

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.
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.
The prime spectrum Spec(R) of a commutative ring R with the Zariski topology is a compact sober space. 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. 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.