So I would like to show that any closed hypersurface (a smooth, compact, and boundaryless manifold of dimension $n$ embedded in $\mathbb{R}^{n+1}$) is orientable. This is an exercise in Guillemin and Pollack's book and the book suggests that the reader use the separation theorem of Jordan and Brouwer.
I understand a fair some of the argument: At each point $p$ in the manifold $M$ consider the tangent space $T_pM$ which can be identified as $\{p\} \times \mathbb{R}^{n} \subset \{p\}\times \mathbb{R}^{n+1}$. We now may assign to each point on the Manifold a normal vector, given by (normalized) vector $v_p$ that spaces the space $\left(\{p\}\times \mathbb{R}^{n}\right)^{\perp}$. Of course, this choice could be multivalued, we might pick $v_p$ or $-v_p$, depending on how we feel, but the point is that we can make a continuous choice of normal vector. To do this, most proofs say something like the following: the Jordan Brouwer theorem allows us to write: $$ \mathbb{R}^{n+1}\setminus M = U_1 \sqcup U_2 $$ where $U_1,U_2$ are both connected components of $\mathbb{R}^{n+1}$ and $U_1$ is bounded whereas $U_2$ is unbounded. It is then said that one should choose at each point the "inward" or "outward" pointing normal (intuitively, the normal whose arrow points towards $U_1$ or the normal whose arrow points towards $U_2$). However, this is seems difficult for me to state rigorously. Does anyone have any ideas as to how one formalizes a normal "pointing inwards" or "pointing outwards"?
