# reference request – Proving that \$C_S^{infty}(M,N)\$ is a Baire space

I have been reading Hirsch’s Differential topology and I am sure a lot of you know this book as a lot of typos. I believe one of them is the proof that $$C_S^{infty}(M,N)$$ is a Baire space. I don’t think his proof works, but I hope this is a true fact since it is used throughout the text. Now when he proves that $$C_S^r(M,N)$$ is a Baire space , for $$0 leq r, we just use a continuous function $$J^r:C^r_S(M,N)rightarrow C^0(M,J^r(M,N))$$ such that the image is a weakly closed subset of $$C^0(M,J^r(M,N))$$ and now since $$J^r(M,N)$$ is complete $$M$$ is a manifold and $$J^r$$ is continuous we get that $$C_S^r(M,N)$$ is a Baire space. Now I think he wants to do the same type of argument for $$J^{infty}:C^{infty}(M,N)rightarrow C^0(M,J^{infty}(M,N))$$ but here we don’t have the fact that the function is continuous. So my question is if anyone knows a proof or a reference for seeing that $$C_S^{infty}(M,N)$$ will in fact be a Baire space? Thanks in advance.