This paper defines a new class of L-fuzzy sets called r-L-fuzzy biconvex sets in (L, M)-fuzzy convex structures (X, C), where C is an (L, M)-fuzzy convexity on X, and some of their properties were studied. In addition, we introduce (L, M)-fuzzy topological convexity space and study some of its properties. Finally, we introduce locally (L, M)-fuzzy topology (L, M)-fuzzy convexity space and study some of its properties,