Index

This illustrates how we can obtain $S^2$ by gluing two copies of
the disc $B^2$ along their boundaries. In more detail, we can put
$$ X = \{(x,y,z)\in\mathbb{R}^3\;|\; x^2+y^2\leq 1 \text{ and } z=\pm 1\} $$
and define $f\colon X\to S^2$ by
$$ f(x,y,z) = (x,y,z\sqrt{1-x^2-y^2}) $$
This is saturated for an appropriate equivalence relation $E$ on
$X$, and the induced map $\overline{f}\colon X/E\to S^2$ is a
homeomorphism.