= Solution
Real line bundles over a paracompact space $X$ are classified by
$$
w_1(L)\in H^1(X;\mathbb F_2).
$$
Euclidean space $\mathbb R^n$ is a <contractible space>, so its first cohomology vanishes and every real line bundle on it is trivial. Apply this to the defining bundle from part (c). In a global trivialization, $s$ is a smooth real function with $Y=s^{-1}(0)$ and $ds|_Y\ne0$. Thus $ds$, or a metric-dual normal vector field, gives a global orientation of the normal line. Combining this with the standard orientation of $\mathbb R^n$ gives an orientation of $Y$ using the same normal-first convention as the <outward-normal-first boundary orientation>. Hence every properly embedded hypersurface in $\mathbb R^n$ is orientable.
Back to article page
