  We prove that the spaces of controlled (branched) rough paths of arbitrary
order form a continuous field of Banach spaces. This structure has many
similarities to an (infinite-dimensional) vector bundle and allows to define a
topology on the total space, the collection of all controlled path spaces,
which turns out to be Polish in the geometric case. The construction is
intrinsic and based on a new approximation result for controlled rough paths.
This framework turns well-known maps such as the rough integration map and the
It\^o-Lyons map into continuous (structure preserving) mappings. Moreover, it
is compatible with previous constructions of interest in the stability theory
for rough integration.
