I assume the reader to be familiar with the classical theory of covering spaces in the topological sense, as in Godbillon [5] or (with some technical differences) in Massey [16], An important branch of the theory concerns the case where the base space is a topological group, which is dealt with more fully in Chevalley [3] but see also Taylor [22], This suggests that it ought to be possible to develop a uniform version of the notion of covering space and in the final section I give an outline of the form which such a theory might take.

In the topological theory there is an essential difference between the notion of local topological equivalence and that of covering map. In the uniform theory we can expect to find we need to strengthen the definition of local uniform equivalence in a similar fashion.

To start with, let R be an equivalence relation on the uniform space X. In Section 2 we have defined the terms weakly compatible and compatible. What we now need is a stronger condition still.