On the closure of one point sets in $T_0$-spaces

Let $X$ be a set and $2^X$ be a set of all subsets of $X$. The necessary and sufficient conditions under which a mapping $X\to 2^X$ is a closure of one-point sets in some $T_0$-space $(X,\tau )$ are described. It is proved that every $T_0$-Alexandroff space is quasi-metrizable by some equidistant quasi-metric.