Definition Diffeo-Topological space [efr-DX6S]

A diffeo-topological space is a diffeological space X equipped with a topology \tau so that all the smooth plots are continuous. A map of diffeo-topological spaces is a smooth map (for the diffeology) which is also continuous (for the topology).

Given a subset X' \subset X, there is an obvious canonical diffeo-topology on X' given by taking the plots to be those functions whose image in X is smooth, and equipping X' with the subspace topology.