Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 1 / 25H / i / Solution

A smooth $n$-dimensional manifold with boundary has charts onto relatively open subsets of the half-space ${\mathbb{H}}^n=\{u_n\ge 0\}$, with smooth transition maps extending smoothly locally across the boundary. A point is a boundary point when its last coordinate in such a chart is zero; this condition is independent of chart. One way to see the independence is that a local diffeomorphism between half-spaces preserves their interior and hence their topological boundary.

Restrict each boundary chart to $u_n=0$ and drop the last coordinate. It maps the boundary onto an open subset of ${\mathbb{R}}^{n-1}$, and the restricted transitions are smooth diffeomorphisms. These charts, with the inherited Hausdorff and countability properties, make $\partial X$ a smooth manifold without boundary. Therefore