(Assuming classical logic) is 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 point.
In each case, half of the definition is that is , the other half states that has enough points:
(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.
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 quotient is sober. Spaces with enough points are also reflective, and a topological space is iff this reflection is sober.
Any nontrivial indiscrete space is not sober, since it is not . More interestingly, the space with the Zariski topology is 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 , which contains “generic points” whose closures are the subvarieties.
The Alexandroff topology on a poset is also not, in general, sober. For instance, if is the infinite binary tree (the set of finite -words (lists) with the “extends” preorder), then the soberification of its Alexandroff topology is Wilson space?, the space of finite or infinite -words (streams).
For instance around definition IX.3. 2 of