Why does Loring Tu define tangent space with equivalent class?

Question

The definition is provided in question: Germs of a $C^{\infty}$ function at $p$, and the derivation at $p$.

I am quite confused that why using “the equivalence class of functions defined in a neighborhood of p and agree on some possibly smaller neighborhood of p”. Why not just defining on all smooth functions? As the section “Definition via derivations” on https://en.m.wikipedia.org/wiki/Tangent_space does

Answer 1

You could define it either way. A priori, you might worry that these are different for two reasons:

1. The action of a derivation on a germ might not be well-defined, i.e., you might have two different functions $f$ and $g$ which agree in a neighborhood of $p$ and a derivation $D$ at $p$ (in the sense of Wikipedia's def, which a priori might be different from Tu's) with $Df \neq Dg$.
2. Maybe you have some function defined near $p$ that doesn't extend to a smooth function on all of $M$. For example, if $M = \mathbb{R}$ and $p=0$, maybe you are looking at the function $\tan(x)$, which is smooth at the neighborhood $(-\pi/2, \pi/2)$ of $p$ but can't be extended to a smooth function on all of $M$. So maybe there are "more" germs near $p$ than global smooth functions, and therefore the derivation condition is satisfied less often.

As you already understand, the question linked above shows that there is nothing to worry about in (1); $Df = Dg$ as soon as $f$ and $g$ agree in a neighborhood of $p$.

As for (2), the point is that there will always be something in the equivalence class of your germ which extends to all of $M$, although you may have to shrink the neighborhood to get it. (For instance, in the given example, restrict the tangent function to $(-\pi/2 + \epsilon, \pi/2 - \epsilon)$ and then extend it to $\mathbb{R}$ by smoothly extending the graph however you like -- you know you can do this in generality thanks to the existence of bump functions and hat functions.)

So the definitions are equivalent. I guess it's reasonable to argue: "why did you make me think about this abstract thing like a germ, if you didn't need it for the definition?" The point is that derivations act a bit more naturally on germs than on global functions, and in particular, this generalizes better to other types of geometric structures.