In this lecture we (finally) get round to defining manifolds with boundary, starting with a topological manifold with boundary and then building from this a smooth manifold with boundary, just as we did in Lecture 1 in the boundary-less case.

In general, my approach to manifolds with boundary will be somewhat cavalier—I don’t want to burden myself with needing to prove everything twice, and thus typically results for manifolds with boundary will be stated without proof. (In fact, in 90% of the course we will have no use for manifolds with boundary).

One area does require careful treatment though, and this is how an orientation of a manifold with boundary $M$ induces an orientation on $\partial M$, since this will be crucial in our treatment of integration next week.