# abstract algebra – Existence of basis on a hairy ball

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?