Reference Request: Whitney $C^k$ topology. I was reading this.
The answer makes use of some facts about $C^k$ topology on smooth maps and differential forms, continuity of the integral operator, etc. Could you suggest references (books) including explanations and proofs of the facts in the answer? Thank you.
 A: As Didier says, there's no need for the Whitney topology there. There are two topologies people tend to call the Whitney topologies, which I will call the weak Whitney (or smooth compact-open) topology and the strong Whitney topology. Each has various advantages.
Some good references are Hirsch's book and the book of Golubitsky and Guillemin for the basics. For something really heady you can take a look at Kriegl-Michor.
Beyond that, most of the time people are interested in what appears to my untrained eye to be global analysis type things, like when $C^{\infty}(M,N)$ acquires the structure of a Banach manifold or something like that. Roughly, as I recall, the idea is to use the exponential map to introduce charts. These results are only worked out when $M$ is compact and $\partial N=\emptyset$, as I recall. I'm not sure if $M$ is permitted to have corners or boundary, and I cannot comment on the other cases. Results of this sort are sadly—as far as I can tell—spread out within various papers within the literature and not yet collected in a single place. This is likely (I'm just spitballing here) because such results weren't needed outside of some highly arcane parts of topology until recently and will perhaps be supplanted (?) by new convenient categories for smooth topology, which is the subject of ongoing research. Anyways, most of this other stuff I've alluded to was worked out in the 70s or earlier and seems to have already taken on the state of folklore status in the 70s. You can find a lot of these papers if you're sufficiently persistent, though.
