CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
A topological space $X$ is sober if its points are exactly determined by its open-set lattice. Different equivalent ways to say this are:
The continuous map from $X$ 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 $X$ the completely prime filters of its open-set lattice is a bijection.
(Assuming classical logic) $X$ is $T_0$ 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 point.
In each case, half of the definition is that $X$ is $T_0$, the other half states that $X$ has enough points:
The continuous map from $X$ 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 $X$ 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.
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, 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 $T_0$ quotient is sober. Spaces with enough points are also reflective, and a topological space is $T_0$ iff this reflection is sober.
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 $Spec(R[x,y])$, which contains “generic points” whose closures are the subvarieties.
The Alexandroff topology on a poset is also not, in general, sober. For instance, if $P$ is the infinite binary tree (the set of finite $\{0,1\}$-words (lists) with the “extends” preorder), then the soberification of its Alexandroff topology is Wilson space?, the space of finite or infinite $\{0,1\}$-words (streams).
For instance around definition IX.3. 2 of