  For a quantale $\mathsf{V}$ we introduce $\mathsf{V}$-approach spaces via
$\mathsf{V}$-valued point-set-distance functions and, when $\mathsf{V}$ is
completely distributive, characterize them in terms of both, so-called closure
towers and ultrafilter convergence relations. When $\mathsf{V}$ is the
two-element chain $\mathsf{2}$, the extended real half-line $[0,\infty]$, or
the quantale ${\bf{\Delta}}$ of distance distribution functions, the general
setting produces known and new results on topological spaces, approach spaces,
and the only recently considered probabilistic approach spaces, as well as on
their functorial interactions with each other.
