# Are these two definitions of smoothness on manifolds equivalent?

I'm wondering if the following very similar definitions of smoothness are equivalent, the first of which is found in Lee's Intro to Smooth Manifolds. Throughout let $$M$$ and $$N$$ be smooth manifolds.

Definition #$$1$$: a map $$F:M\to N$$ is smooth iff for every $$p\in M$$ there exist charts $$(U,\phi)$$ and $$(V,\psi)$$ containing $$p$$ and $$F(p)$$ respectively such that

• $$F(U)\subseteq V$$, and
• $$\psi\circ F\circ\phi^{-1}:\phi(U)\to\psi(F(U))$$ is smooth.

Definition #$$2$$: a map $$F:M\to N$$ is smooth iff for every $$p\in M$$ there exist charts $$(U,\phi)$$ and $$(V,\psi)$$ containing $$p$$ and $$F(p)$$ respectively such that

• $$\psi\circ F\circ\phi^{-1}:\phi(U)\to\psi(F(U)\cap V)$$ is smooth.

Equivalence: naturally the first definition implies the second one, but I'm struggling to show the converse. Assuming the hypothesis in the second definition holds, it seems it would be sufficient to find charts $$(U_*,\phi_*)$$ and $$(V_*,\psi_*)$$ of $$p$$ and $$F(p)$$ respectively such that

• $$U_*\subseteq F^{-1}(F(U)\cap V)$$, and
• $$V_*\supseteq F(U_*)$$,

since then the map $$\psi_*\circ F\circ \phi^{-1}_*:\phi_*(U_*)\to\psi_*(F(U_*))$$ is smooth as it can be decomposed as $$(\psi_*\circ\psi^{-1})\circ(\psi\circ F\circ\phi^{-1})\circ(\phi\circ\phi^{-1}_*)$$ where each expression in parenthesis is smooth.

• Where did you get the second definition? $F$ need not map $U$ into $V$, so as is, the codomain of the map $\phi(U)\to \psi(F(U)\cap V)$ makes it not well defined. It would be better to write $\psi\circ F\circ \phi^{-1}: \phi(F^{-1}(F(U)\cap V)\cap U)\to \psi(F(U)\cap V)$. Commented Aug 14, 2023 at 19:26

As J. V. Gaiter noted in a comment, Definition #2 doesn't make sense as written, because the composite map $$\psi\circ F \circ \phi^{-1}$$ might not be defined on all of $$\phi(U)$$.
If you modify the definition the way J. V. Gaiter suggested, or equivalently and more simply just writing $$\psi\circ F \circ \phi^{-1} \colon \phi\big(F^{-1}(V) \cap U\big) \to \psi(V),$$ then the problem is that the domain of the composite map need not be open, so you have to decide what you mean by "smoothness" for a map whose domain is not an open subset of $$\mathbb R^n$$. The usual definition is that such a map is smooth if every point of the domain has an open neighborhood on which the given map is the restriction of a smooth map in the usual sense. But with this understanding, Definition #2 is not equivalent to Definition #1. I described a counterexample in Problem 2-1 of Introduction to Smooth Manifolds: If $$f\colon \mathbb R\to \mathbb R$$ is the function $$f(x) = \begin{cases} 1, & x\ge 0,\\ 0, & x < 0, \end{cases}$$ then for every $$x\in \mathbb R$$ there are smooth charts $$(U,\phi)$$ containing $$x$$ and $$(V,\psi)$$ containing $$f(x)$$ such that $$\psi\circ f \circ \phi^{-1}$$ is smooth as a map from $$\phi(f^{-1}(V)\cap U)$$ to $$\psi(V)$$ in the sense described above, but $$f$$ is not continuous.
If you add to Definition #2 the hypothesis that $$F$$ is continuous (in addition to correcting the domain as mentioned above), then you get a definition equivalent to #1. See Proposition 2.5 in ISM.