I'm reading Tu's "An Introduction to Manifolds", and he defines the tangent bundle on $M$ as the disjoint union $TM:=\bigcup_{p\in M}\{p\}\times T_pM$, but he remarks that for $p\neq q$, we already have $T_pM\cap T_qM=\emptyset$. Pictorially, I get what he's saying (a vector at $p$ can't also be a vector at $q$), but with the formal definition using derivations, I can't see why this is necessarily so.
$T_pM$ is defined as the set of derivations $X_p:C_p^{\infty}(M)\to\mathbb{R}$, where $[f]$ in $C_p^{\infty}(M)$ is the set $\{g\in C^{\infty}(M): p\in (U,\phi)\in{\scr A}, f|_U=g|_U \}$. I think Tu is saying that since $C_p^{\infty}(M)\neq C_q^{\infty}(M)$, the domains of the derivations $X_p$ and $X_q$ are different, hence $X_p\neq X_q$. But is it always true that $C_p^{\infty}(M)\neq C_q^{\infty}(M)$? For example, if $M$ is homeomorphic to an open subset of $\mathbb{R}^n$, then we have an atlas $\scr A$ of $M$ with only one chart $(M,\phi)$, and so we would have to have $C_p^{\infty}(M)=C_q^{\infty}(M)$, right?