I'll be quoting from the Wikipedia page on smoothness. Smooth function between manifolds are defined as follows:
If $F$ is a map from an $m$-manifold $M$ to an $n$-manifold $N$, then $F$ is smooth if, for every $p\in M$, there is a chart $(U, \varphi)$ in $M$ containing $p$ and a chart $(V, \psi)$ in $N$ containing $F(p)$ with $F(U) \subset V$, such that $\psi\circ F \circ \varphi^{-1}$ is smooth from $\varphi(U)$ to $\psi(V)$ as a function from $\mathbb R^m$ to $\mathbb R^n$.
They then define a notion of a smooth map between arbitrary subsets of manifolds:
If $f\colon X \to Y$ is a function whose domain and range are subsets of manifolds $X \subset M$ and $Y \subset N$, respectively, $f$ is said to be smooth if for all $x\in X$ there is an open set $U\subset M$ with $x\in U$ and a smooth function $F\colon U\to N$ such that $F(p) = f(p)$ for all $p \in U \cap X$.
The funny thing is, they never defined a smooth map on an open subset of a manifold, even though they use this notion in their definition of smooth maps between arbitrary subset of manifolds! So my question is, how are we supposed to define a smooth map on an open subset of a manifold? Are we supposed to give the open set the structure of a manifold by giving it the subspace topology and forming an atlas on it from the set of all charts in the initial atlas contained in the open subset?
Best Answer
Let $M$ be a topological space and $U$ be a subset of $M$. If you want the set-theoretical inclusion map $\iota : U \longrightarrow M$ to be continuous, the subspace topology is the smallest one you need. In the case of the manifold and its open set, if you want $\iota$ to be a morphism in this category ($C^0$, $C^r$, $C^\infty$, complex, etc.), the extra condition you will find is that the atlas of $M$ and that of $U$ are "compatible". Note you cannot just collect all the members of the atlas which are contained in $U$ to form the atlas of $U$. Those charts may simply be too large. Take $S^n$ with only two charts as an example. The canonical way is to collect all the pairs of the form $(U \cap V, \varphi \big|_U)$ to form the atlas, where $(V, \varphi)$ is a chart of $M$.