Consider the standard solid ball $\{(x,y,z)\in \mathbb{R}^3\mid x^2+y^2+z^2\le 1\}$, and the equivalence relation doing nothing in the interior and identifying antipodal points of the boundary. What does the quotient space look like? Is it a manifold? What does its boundary (if it has one) look like?
It seems to me that the equivalence relation can be related to that generating the real projective plane $P^2(\mathbb R)$; in that case the action is free and properly discontinuous, so that the quotient is a manifold. But it this case, it seems I built something whose boundary is $P^2(\mathbb R)$, a well-known contradiction.
Where am I wrong?