The set of isomorphisms of a topological space to itself form a group. The topology of a topological group is determined by the neighborhoods of the group identity. The problem is saying what is a small transformation. If the space were metric we could say that f is small if for all x in the space δ(x, f(x)) < 0.001 . Lacking a metric this is meaningless. It would seem that we need a uniformity.

Barden’s “An Introduction to Differential Manifolds” suggests that the ‘compact-open’ topology is a suitable topology for the set of continuous permutations of the fiber.