Given two topological spaces the idea of the Cartesian product is natural and there is only one good idea for the topology for the product. If X and Y are topological spaces then there there are two two general topologies for XY. (XY is the set of continuous functions from Y to X.) XX is the set of endomorphisms of a topological space X.

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.