Question about the definition of smooth map between manifolds

Question

$F:M \rightarrow N$ is smooth if and only if $F: M \rightarrow N$ is continuous and for every chart $(U, \phi)$ on M, $(V, \psi)$ on N, then $ \psi \circ F \circ \phi^{-1}:\phi (U\cap F^{-1}(V)) \rightarrow \psi(V)$ is smooth. My question is: Let $a \in U \cap F^{-1}(V) \Rightarrow a \in U$ and $F(a) \in V$. Then $\phi (a) \in \phi(U)$, and $\psi (F (\phi^{-1} (\phi(a)))= \psi(F(a))$ since $\phi $ is homeomorphism. $F(a) \in V$, and $\psi: V \rightarrow \psi(V)$ is a chart, so it is smooth, so $\psi (F(a))$ is smooth. Then every map is smooth?

Answer 1

Note that a chart being smooth doesn't make sense (a priori, unless you regard euclidean space as a smooth manifold itself). And points are also not smooth, so I guess you meant $\psi\circ F$ is smooth. But again this doesn't make sense a priori (unless regarding euclidean spaces as smooth manifolds). And even a in that case, this may not be true if $F$ is not smooth to begin with.

It is important to note that smooth is a notion that initially only makes sense for maps between open sets of euclidean spaces. The way we extend this notion to the notion of smooth map between manifolds is with the definition that you state at the beginning of the question. So always ask yourself what does every word mean when you say it. This is just a bit confusing, because smooth had an initial meaning and now we use the word smooth for something else a bit more abstract. But as I said before, the usual smooth maps between euclidean spaces are also naturally smooth in this sense, and this is why your (potentially confusing) statement "charts are smooth" is technically correct.