In this paper, the notion of (L, M)-fuzzy topological-convex spaces is introduced and some of its characterizations are obtained. Then the notion of (L, M)-fuzzy convex enclosed relation spaces is introduced and its one-to-one correspondence with (L, M)-fuzzy convex space is studied. Based on this, the notion of (L, M)-fuzzy topological-convex enclosed relation spaces is introduced and its categorical isomorphism to (L, M)-fuzzy topological-convex spaces is discussed.

In this paper, axiomatic definitions of both L-convex bases and L-convex subbases are introduced and their relations with L-convex spaces are studied. Based on this, the notion of L-topological-convex space is introduced as a triple (X, 𝓣, 𝓒), where X is a nonempty set, 𝓒 is an L-convex structure on X and 𝓣 is an L-cotopology on X compatible with 𝓒. It can be characterized by many means.

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