Sufficiently near L-fuzzy sets

In this paper, we introduce the category LεAMer having objects that are Lε-approach merotopic spaces, where ε ∈ (0, ∞]. In addition, we consider an Lε-approach merotopy that provides a means of measuring the almost nearness of pairs of collections of L-fuzzy sets. The order structure of a family of Lε-approach merotopies on a nonempty set is also taken into consideration. Examples are given to support the existence of such structures.