What is the intersection of the positively oriented $n$-covectors with the fiber of the $n$-covectors at each point of a manifold? In the below proof of Proposition 15.5 from John Lee's Introduction to Smooth Manifolds, I cannot figure out the second sentence. Why is the intersection of the positively oriented $n$-covectors with the fiber of the $n$-covector at $p$, an open half line?


 A: An orientation on $M$ is the datum of an oriented atlas $\{(U_{\alpha},\varphi_{\alpha})\}$, that is an atlas with $\det (\mathrm{d} (\varphi_{\alpha}\circ \varphi_{\beta}^{-1}))>0$ for all $\alpha,\beta$ with $U_{\alpha}\cap U_{\beta} \neq \varnothing$.
Write $\varphi_{\alpha} = (x^1_{\alpha},\ldots,x^n_{\alpha})$ as a coordinate system, and $({\partial_1}_{\alpha},\ldots,{\partial_n}_{\alpha})$ its tangent frame. The atlas being oriented hence means that at each point $p \in U_{\alpha}\cap U_{\beta}$, the two bases $({\partial_1}_{\alpha}|_p,\ldots,{\partial_n}_{\alpha}|_p)$ and $({\partial_1}_{\beta}|_p,\ldots,{\partial_n}_{\beta}|_p)$ define the same orientation of $T_pM$: with calculus, this reads
$$
\det_{({\partial_1}_{\alpha}|_p,\ldots,{\partial_n}_{\alpha}|_p)}({\partial_1}_{\beta}|_p,\ldots,{\partial_n}_{\beta}|_p) >0
$$
A positively oriented $n$-covector $\omega_p$ is an element $\omega_p \in \Lambda^n(T_pM^*)$ such that
$$
\omega_p(\partial_1|_p,\ldots,\partial_n|_p) >0
$$
which in fact does not depend on the choice of the coordinate patch of the oriented atlas since
$$
\omega_p({\partial_1}_{\beta}|_p,\ldots,{\partial_n}_{\beta}|_p) = \omega_p({\partial_1}_{\alpha}|_p,\ldots,{\partial_n}_{\alpha}|_p) \times \det_{({\partial_1}_{\alpha}|_p,\ldots,{\partial_n}_{\alpha}|_p)}({\partial_1}_{\beta}|_p,\ldots,{\partial_n}_{\beta}|_p)
$$
(to be honest, I did not check if the determinant has to be on the LHS or the RHS, and it does not matter: the only thing that matters is that the LHS and RHS have same sign).
Now, recall that if $V$ is a $d$-dimensional vector space, then $\Lambda^dV$ is a $d \choose d$-dimensional vector space, that is, a line. Therefore, $\Lambda^n(T_pM^*)$ is a line. If $\omega_p \in \Lambda^n(T_pM^*) \setminus \{0\}$, then either $\omega_p(\partial_1|_p,\ldots,\partial_n|_p) >0$ for any oriented coordinate patch or $\omega_p(\partial_1|_p,\ldots,\partial_n|_p) <0$ for any oriented coordinate patch. Geometrically speaking, we have cut the line $\Lambda^n(T_pM^*)$ at $0$, and the rest consists in two half lines, one half whose elements are positive, the other half whose elements are negative.
