The normality of L-coarse spaces

For L being a completely distributive lattice, we introduce a notion of L-coarse spaces, which can be regarded as a large-scale analogue of probabilistic uniform spaces. The main goal of this paper is to investigate the properties of L-coarse spaces that can induce L-valued coarse proximity spaces. First of all, we introduce a notion of L-valued coarse neighborhood systems, and show that the resulting category is isomorphic to that of L-valued coarse proximity spaces. Then we study the normality of coarsely connected L-coarse spaces and of asymptotically connected L-valued asymptotic resemblance spaces, and show that these two normality properties are coincident. More importantly, we show that a coarsely connected L-coarse space induces an L-valued coarse proximity if and only if it is coarsely normal. We conclude with noting that the previous result are also valid for asymptotically connected L-valued asymptotic resemblance spaces.