Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 115 / 1 / d / Solution

Real line bundles over a paracompact space $X$ are classified byEuclidean 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 $ds{|}_Y\ne 0$. 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.