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.