Suppose $A \subset M$ is a subset of a smooth manifold $M$ and $A$ is also a smooth manifold with some arbitrary topology and smooth structure. $A$ is therefore just some arbitrary subset that also turns out to be a smooth manifold.
It seems to me that the inclusion $i : A \to M$ is then a smooth map as its restriction on any open subset $U \subset A$ equals the identity on the open submanifold $U$ in $A$: $$ i\big|_U = \text{id}_U : U \to U, $$ which is smooth. I am following John Lee in his introduction to smooth manifolds, but he seems to state the result only if $A$ is a submanifold of some kind. Could you clarify if I am right or wrong? And in the case I am wrong, please explain why.
The usual definition of a smooth map of manifolds $f : M \to N$ depends on charts on $M$ and $N$. If your subset has no charts, it does not seem to me that you can properly talk about smoothness.