I'm currently reading Introduction to Smooth Manifolds by John M. lee. I'm trying to understand his definition of smooth maps $F:A\subseteq M\to N$ given in page 45. Let's start from scratch to have some context.
Definition 1. If $A\subseteq \mathbb{R}^n$ is an arbitrary subset, a function $F:A\to \mathbb{R}^m$ is said to be smooth on $A$ if it admits a smooth extension to an open neighborhood of each point, or more precisely, if for every $x\in A$, there exist an open subset $U_x\subseteq \mathbb{R}^n$ containing $x$ and a smooth function $\tilde F:U_x\to \mathbb{R^m}$ that agrees with $F$ on $U_x\cap A$. The notion of diffeomorphism extends to arbitrary subsets in the obvious way: given arbitrary subsets $A$, $B\subseteq \mathbb{R}^n$, a diffeomorphism form $A$ to $B$ is smooth bijective map $f:A\to B$ with smooth inverse.
It's easy to see that this defintion of smoothness agrees with the usual one if $A$ is an open subset of $\mathbb{R}^n$.
Definition 2. Let $M$ be a topological manifold with boundary, i.e. a second countable hausdorff space such that each point $p\in M$ has a neighborhood homeomorphic to an open subset of $\mathbb{R}^n$ or $\mathbb{H}^n$. A smooth structure for $M$ is defined to be a maximal smooth atlas-a collection of charts whose domains cover $M$ and whose transition maps (and their inverses) are smooth like in Definition 1. Every smooth manifold is automatically a smooth manifold with boundary (whose boundary is empty).
Everything is clear there but this is where things get tricky, in the book Lee defines smoothness of functions with arbitrary domains as follow:
Definition 3. Suppose $M$ and $N$ are smooth manifolds with or without boundary and $A\subseteq M$ is an arbitrary subset. We say that a map $F:A\to N$ is smooth on $A$ if it has a smooth extension in a neighborhood of each point: that is, if for every $p\in A$ there is an open subset $W\subseteq M$ containing $p$ and a smooth map $\tilde F:W\to N$ whose restriction to $W\cap A$ agrees with $F$.
However, this definition is wrong. Lee himself pointed that out in the errata on his webpage and here (Lee's) Definition of smooth maps on manifolds, in particular vector fields.
In the errata, he proposes to change the definition to the following one:
Definition 4. If $N$ has empty boundary, we say that a map $F:A\to N$ is smooth on $A$ if it has a smooth extension in a neighborhood of each point: that is, if for every $p\in A$ there exist an open subset $W\subseteq M$ containing $p$ and a smooth map $\tilde F:W\to N$ whose restriction to $W\cap A$ agrees with $F$. When $\partial N\neq \emptyset$, we say $F:A\to N$ is smooth on $A$ if for every $p\in A$ there exist an open subset $W\subseteq M$ containing $p$ and a smooth chart $(V,\psi)$ for $N$ whose domain contains $F(p)$, such that $F(W\cap A)\subseteq V$ and $\psi \circ F|W\cap A$ is smooth as a map into $\mathbb{R}^n$ like in Definition 3.
The new definition fixes the problem of the definition given in the book. However there is something I dislike about the new definition: It first defines the concept when $\partial N=\emptyset$ and then defines it when $\partial N$ might not be empty. Of course, the definition given in the book and the new definition agree when $\partial N=\emptyset$, however this way of doing things adds a new layer of complication. If we were very strict, the new defintion (where it doesn't matter whether $\partial N$ is empty or not) would be like this:
Defintion 4(b) Let $M,N$ be smooth manifolds with boundary and $A\subseteq M$. A map $F:A\to N$ is smooth on $A$ if for each $p\in A$ there is a neighborhood $W$ of $p$, a smooth chart $(V,\psi)$ for $N$ such that $F(W\cap A)\subseteq V$ and $\psi\circ F|W\cap A$ is smooth. $\psi \circ F|W\cap A$ smooth means that for every $q\in W\cap A$ there is $U$ neighborhood of $q$ contained in $W$ and a smooth function $\tilde F:U\to \mathbb{R}^n$ that agrees with $\psi \circ F|W\cap A$ in $U$.
That definition looks quite complicated but that is the one implied by Lee in the errata. The problem is that it relies on a definition given when $\partial N\neq \emptyset$. The definition of a smooth structure on a manifold with boundary doesn't have this problem because it deals with both cases (smooth manifolds and smooth manifolds with boundary) at the same time. The point is that the Definition 1. of smoothness agrees with the old one, so we can treat both cases simultaneously.
So my question is: Is there a way to simplify the definition given by Lee in the errata? More concretely, is there a way to deal with both cases (smooth manifolds and smooth manifolds with boundary) simultaneously?
This really isn't part of the question but I'll post it here because I think it's worthy. Why not simplify Definition 1. by saying $F:A\to \mathbb{R}^n$ is smooth if there is an open set $U$ containing $A$ and a smooth function $\tilde F:U\to \mathbb{R}^n$ such that $\tilde F$ agrees with $F$ on $A$. This alternative definition and Definition 1. are equivalent by using partitions of unity.
Best Answer
Unfortunately Prof. Lee hasn't answered but I think I arrived at a satisfactory answer. There is a way to treat smooth manifolds and smooth manifolds with boundary simultaneously, however when the codomain is a smooth manifold the situation is slightly simpler.
The following paragraphs justify the given definitions.
Let's prove that the general definition matches the simplified definition when $N$ is a smooth manifold (without boundary).
Simplified definition. $\implies$ General definition. Let $p\in A$, $\tilde W$ neighborhood of $p$ and $\tilde F:\tilde W\to N$ smooth such that $\tilde F|\tilde W\cap A=F|\tilde W\cap A$. Let $(V,\psi)$ be any chart for $N$ containing $f(p)$. Define $W=F^{-1}(V)$, then $\psi \circ \tilde F|W:W\to R^n$ is smooth because $\tilde F|W:W\to V$ and $\psi:V\to R^n$ are smooth. Also $F(W\cap A)=\tilde F(W\cap A)\subseteq V$. Finally $(\psi \circ \tilde F|W)|W\cap A=\psi \circ F|W\cap A$.
General definition. $\implies$ Simplified definition. Let $p\in A$, $W$ neighborhood of $p$, $(V,\psi)$ chart for $N$ and $\tilde F:W\to R^n$ smooth such that $F(W\cap A)\subseteq V$ and $\tilde F|W\cap A=\psi \circ F|W\cap A$. Define $\tilde W=\tilde F^{-1}(\psi(V))$, note that $\tilde W$ is a neighborhood of $p$ because $\tilde F(p)=\psi(F(p))\in \psi (V)$ and $\psi(V)$ is an open subset of $R^n$ because $N$ is a smooth manifold (without boundary). Now $\psi ^{-1}\circ \tilde F|\tilde W:\tilde W\to V\subseteq N$ is smooth because is the composition of $\psi^{-1}$ and $\tilde F|\tilde W$. Finally $(\psi^{-1}\circ \tilde F|\tilde W)|\tilde W\cap A=F|\tilde W\cap A$.
Let's prove that the general definition matches Definition 4(b).
Definition 4(b) $\implies$ General definition. Let $p\in A$, $\tilde W$ neighborhood of $p$ and $(V,\psi)$ chart for $N$ such that $F(\tilde W\cap A)\subseteq V$ and $\psi\circ F|\tilde W\cap A:\tilde W\cap A\to R^n$ is smooth as in Definition 3. Particularly there is a neighborhood $W$ of $p$ and a smooth map $\tilde F:W\to R^n$ with $W\subseteq \tilde W$ and $\tilde F|W\cap A=\psi \circ F|W\cap A$ ,i.e. pick $q=p$ in the notation of Definition 4(b). Finally $F(W\cap A)=\psi^{-1}(\tilde F(W\cap A))\subseteq \psi^{-1}(\psi(V))=V$.
General definition. $\implies$ Definition 4(b) Let $p\in A$, $W$ neighborhood of $p$, $(V,\psi)$ chart for $N$ and $\tilde F:W\to R^n$ smooth such that $F(W\cap A)\subseteq V$ and $\tilde F|W\cap A=\psi \circ F|W\cap A$. Let $q\in W\cap A$ (as in Definition 4(b)) and define $U=W$.