By a closed class of operations we mean a set of finitary operations on a set A closed with respect to composition of operations and identification and permutation of variables in operations. Closed classes ordered by inclusion form a complete algebraic lattice 5e. If A is finite, the lattice 5 ° is dually atomic and the dual atoms, called maximal or precomplete classes, play an important role in primality (i.e. functional completeness) criteria [2, 8, 12, 13, 15]. For infinite A's a few maximal classes in the countable case were determined by Gavrilov. In this paper some of his results are extended to arbitrary infinite sets A. Gavrilov [5, 6] in the countable case and the author for the general case have shown indirectly that there are as many maximal classes as there are closed classes (i.e. 22~A~). The proofs of this fact seems to indicate that the structure of the maximal classes is rather complex. It is also not known whether ~ is dually atomic. Although there are no practical reasons for the investigation of the maximal classes on infinite sets, the problem seems to be interesting enough in itself to warrant this small excursion into a largely unknown area of universal algebra.