# Is the tangent bundle the DISJOINT union of tangent spaces?

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.

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How did Lee and Tu respectively define $T_pM$? – Neal Nov 4 '12 at 12:32
This should also be differential-topology, not differential-geometry. – M.B. Nov 4 '12 at 12:33

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".
Tu's definition of derivation is different from that in Lee's book. The first, uses germs instead the latter says that a derivation is a linear function with domain $C ^{\infty} (M)$ – Dubious Nov 4 '12 at 15:58
As an algebraic/analytic geometer I much prefer the definition with germs. The other definition works in differential geometry because of the specific result that the canonical map $\mathcal C^\infty(M) \to \mathcal C^\infty_p$ is surjective. This is completely false in the algebraic/analytic category. – Georges Elencwajg Nov 4 '12 at 17:20