Let $M$ be a smooth manifold and consider the Lee's definition of the tangent space $T_pM$ (so $T_pM$ is the vector space of derivations at $p$). The canonical definition of tangent bundle (as set) of $M$ is:
$$TM=\bigcup_{p\in M}\{ p\}\times T_pM$$
so it is the disjoint union of all tangent spaces; but L.W.Tu in his "Introduction to Manifolds" says that the tangent spaces are already disjoint and for this reason he defines
$$TM=\bigcup_{p\in M} T_pM$$
Why we can't find a common derivation between $T_pM$ and $T_qM$ if $q\neq p$? I think that Tu's statement is not true.
Best Answer
A derivation at $p\in M$ is in particular a linear map $\partial: C^\infty_p \to \mathbb R$ defined on the set of germs of smooth functions at $p$, and similarly for $q$.
So the sets of derivations at $p$ and $q$ are disjoint simply because they consist of maps with different domains (namely $C^\infty_p$ and $C^\infty_q$). And maps with different domains cannot be equal, as follows from the set-theoretical definition of "map".