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 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.

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 from its partially ordered set of open subsets to $\$ that preserves all joins and all finite meets 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 $x_$ is ''self-convergent'' if it converges to every point $x_i$ in $x_$, or equivalently if its eventuality filter is completely prime. A net $x_$ 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 $x_$ 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 (T₂) space is sober (the only irreducible subsets being singletons), and all sober spaces are Kolmogorov (T₀), and both implications are strict.

Sobriety is not comparable to the T₁ condition:
* an example of a T₁ space that 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 that is not T₁ is the Sierpinski space.

Moreover, T₂ is strictly stronger than T₁ ''and'' sober, i.e., while every T₂ space is at once T₁ and sober, there exist spaces that are simultaneously T₁ and sober, but not T₂. 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 T₀ 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.

See also

* Stone duality, on the duality between topological spaces that are sober and frames (i.e. complete Heyting algebras) that are spatial.

References

Further reading

*
* {{cite book , last=Vickers , first=Steven , authorlink=Steve Vickers (computer scientist) , title=Topology via logic , series=Cambridge Tracts in Theoretical Computer Science , volume=5 , location=Cambridge , publisher=Cambridge University Press , year=1989 , isbn=0-521-36062-5 , zbl=0668.54001 , page=66

General topology
Separation axioms
Properties of topological spaces