Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 37D / a

The antipodal action fixes the origin. Away from the origin it acts freely, so the quotient is locally Euclidean there. A small punctured neighborhood of the image of the origin, however, has linkthe real projective plane, whereas the link of a point in a three-dimensional smooth manifold is $S^2$. Equivalently, the neighborhood is a cone on ${\mathbb{RP}}^2$ and cannot be a three-ball. Thus

Removing the closed unit ball leaveswhich is an open subset of ${\mathbb{R}}^3$. Every point therefore has an ordinary Euclidean coordinate neighborhood, soIt has no boundary; the unit sphere was removed together with the ball.