nLab
sober space

Definition

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

• $X$ is isomorphic to the space of points of the locale it gives rise to.

• The points of $X$ are in bijection with the completely prime filters of its open-set lattice.

• (Assuming classical logic) $X$ is $T_0$ and every irreducible closed set (closed set that is not the union of any two smaller closed sets) is the closure of a point.

Sobriety is a separation property that is stronger than $T_0$, but incomparable with $T_1$. 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.

Examples

Any nontrivial indiscrete space is not sober, since it is not $T_0$. More interestingly, the space $R^2$ with the Zariski topology? is $T_1$ 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 $\mathrm{Spec}(R[x,y])$, which contains “generic points” whose closures are the subvarieties.

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