CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
If $\mathcal{O}(X)$ is the topology on a topological space $X$, and if a map $\mathcal{O}(X) \to \mathcal{O}(1)$ that preserves finite meets and arbitrary joins is considered an instance of â€śseeing a point $1 \to X$â€ť, then $X$ is â€śsoberâ€ť if every point we see is really there (i.e., is induced from a point = continuous map $1 \to X$), and if we never see double.
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 $X$ to 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 (unique, by $T_0$) 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.
The following non-example shows that sobriety is not a hereditary separation property, i.e., subspaces of sober spaces need not be sober:
For instance around definition IX.3.2 of