Fuzzy maps are studied by stressing the idea that they are respectful with granularity when it is modeled by the existence of fuzzy equivalence relations (indistinguishability operators). The granules of the system are then identified with their fuzzy points. Some properties of perfect and classical fuzzy maps are stated and the existence of maximal fuzzy maps is proved. Maximal fuzzy maps handle the greatest uncertainty, in the sense that the granules are of the greatest possible size, so they are the less specific maps in the system. Since granularity is expressed with fuzzy points, they are also studied in the paper. The existence of maximal fuzzy points is established and they are characterized as the fixed points of a special map that we call .

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