This paper presents the concepts of (L, M)-remotehood spaces and (L, M)-convergence spaces in the framework of (L, M)-fuzzy convex spaces. Firstly, it is shown that the category of (L, M)-remotehood spaces is isomorphic to the category of (L, M)-fuzzy convex spaces. Secondly, it is proved that the category of (L, M)-fuzzy convex spaces can be embedded in the category of (L, M)-convergence spaces as a reflective subcategory. Finally, the concepts of preconvex (L, M)-remotehood spaces and preconvex (L, M)-convergence spaces are introduced and it is shown that the category of preconvex (L, M)-remotehood spaces is isomorphic to the category of preconvex (L, M)-convergence spaces.