In this section we work over a uniform base space B. By a uniform space over B I mean a uniform space X together with a uniformly continuous function p : X → B, called the projection. Usually X alone is sufficient notation. Thus B can always be regarded as a uniform space over itself, with the identity as projection. Also the uniform product B × T can be regarded as a uniform space over B for any uniform space T, using the first projection. Again, if G is a topological group acting uniformly equicontinuously on the uniform space X through uniform equivalences, then X can be regarded as a uniform space over X/G, with the quotient uniform structure, using the natural projection.

If X is a uniform space over B with projection p the preimage p−1 (B') of a subspace B' of B is usually denoted by XB', and may be regarded as a uniform space over B' by restriction of the projection. When B' is a point-space b, say, the preimage is called the fibre and denoted by Xb.