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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 $T_pM$?2012-11-04
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    This should also be differential-topology, not differential-geometry.2012-11-04
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    I just noticed this question. I'll add that with the definition of $T_pM$ that I give in my book (derivations of $C^\infty(M)$ at $p$), the zero derivation is a derivation at $p$ for every $p$, so defining the tangent bundle as a simple union would not work. I understand the advantages of defining $T_pM$ as the set of derivations of the space of germs (and I often think of it that way myself), but for pedagogical reasons I made the decision to use the conceptually simpler definition involving $C^\infty(M)$, which I thought would be a little easier for novices to wrap their heads around.2017-06-09
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    @JackLee https://math.stackexchange.com/questions/2982860/why-are-c-infty-p-neq-c-infty-q-when-p-neq-q2018-11-03

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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".

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    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)$2012-11-04
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    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.2012-11-04
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    But how are we sure that we always have $C^\infty_p \neq C^\infty_q$ for $q\neq p$?2018-11-03