nLab
sober topological space

Contents

Idea

If 𝒪(X)\mathcal{O}(X) is the topology on a topological space XX, and if a map 𝒪(X)𝒪(1)\mathcal{O}(X) \to \mathcal{O}(1) that preserves finite meets and arbitrary joins is considered an instance of “seeing a point 1X1 \to X”, then XX is “sober” if every point we see is really there (i.e., is induced from a point = continuous map 1X1 \to X), and if we never see double.

Definition

A topological space XX is sober if its points are exactly determined by its open-set lattice. Different equivalent ways to say this are:

  • The continuous map from XX to the space of points of the locale that it gives rise to (see there for details) is a homeomorphism.

  • The function from points of XX the completely prime filters of its open-set lattice is a bijection.

  • (Assuming classical logic) XX is T 0T_0 and every irreducible closed set (non-empty closed set that is not the union of any two non-empty closed sets) is the closure of a (unique, by T 0T_0) point.

In each case, half of the definition is that XX is T 0T_0, the other half states that XX has enough points:

  • The continuous map from XX to the space of points of the locale that it gives rise to (see there for details) is a quotient map.

  • The function from points of XX the completely prime filters of its open-set lattice is a surjection.

  • (Assuming classical logic) every irreducible closed set (non-empty closed set that is not the union of any two non-empty closed sets) is the closure of a point.

Properties

  • Sobriety is a separation property that is stronger than T 0T_0, but incomparable with T 1T_1. With classical logic, every Hausdorff space is sober, but this can fail constructively.

  • The category of sober spaces is reflective in the category of all topological spaces; the left adjoint is called the soberification. This reflection is also induced by the idempotent adjunction between spaces and locales; thus sober spaces are precisely those spaces that are the space of points of some locale, and the category of sober spaces is equivalent to the category of locales with enough points.

  • A topological space has enough points if and only if its T 0T_0 quotient is sober. Spaces with enough points are also reflective, and a topological space is T 0T_0 iff this reflection is sober.

Examples (and non-examples)

  • Any nontrivial indiscrete space is not sober, since it is not T 0T_0. More interestingly, the space R 2R^2 with the Zariski topology is T 1T_1 but not sober, since every subvariety is an irreducible closed set which is not the closure of a point. Its soberification is, unsurprisingly, the scheme Spec(R[x,y])Spec(R[x,y]), which contains “generic points” whose closures are the subvarieties.

The last (non-)example shows that sobriety is not a hereditary separation property, i.e., subspaces of sober spaces need not be sober.

  • The Alexandroff topology on a poset is also not, in general, sober. For instance, if PP is the infinite binary tree (the set of finite {0,1}\{0,1\}-words (lists) with the “extends” preorder), then the soberification of its Alexandroff topology is Wilson space?, the space of finite or infinite {0,1}\{0,1\}-words (streams).

References

For instance around definition IX.3.2 of

Revised on October 29, 2014 20:24:27 by Bas Spitters (83.248.192.75)