Quantale-valued Approach Spaces via Closure and Convergence

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.