I am watching, lecture 6 of “The Mathematics and Physics of Gravity and Light” where the lecturer brings up the case of hairy ball to show that $Γ(TS^2)$ $C^∞$-module (a vector space of smooth sections over a $C^∞$ ring) does not have a basis. I have read a couple other posts regarding this, but I am still confused. Specifically, in order for the above cases to prove that $C^∞$-module does not have a basis, then it implicitly assumes that there exists a basis in $Γ(TS^2)$ $mathbb{R}$-vector space which is a vector space over a field of real numbers.

Now, if I’ve understood correctly, an element in $Γ(TS^2)$ $C^∞$-module allows scaling of a vector field at individual points. Thus linear combination of them can give a vector field that has curl. Thus $Γ(TS^2)$ $C^∞$-module having no basis in hairy ball seems to amount to saying that a hairy ball cannot be combed in a single smooth stroke.

However, it also seems impossible that the elements of $Γ(TS^2)$ $mathbb{R}$-vector which as I understood should be a vector field with vectors at each point pointing in a same direction scaled equally by $repsilonmathbb{R}$, to have a basis. So am I misunderstanding something? If not, if the hairy ball example seems to show non-existence of basis for both a vector space and a module, how does it illustrate the non-existence of basis on a module?