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?