I am reading a book on homological algebra. In order to determine the homology group of the $2$-dimensional projective space, the author identifies the space with the southern hemisphere of $S^2$, together with $s/\{ \pm 1 \}$, $s$ denoting the equator. $\mathbb{PR}^2 = \mathbb{R}^2 \cup \ell_{\infty}$.
That is:
$P \in \mathbb{R}^2$ is identified with $P' \in S^2$. $A$ and $B$ on $s$ represents the same point in $\mathbb{PR}^2$.
Then, this hemisphere is homeomorphic to a disk, and this disk is transformed continuously to a rectangular:
And this can be simplified to:
I can understand the first two parts, but I don't know how does the simplification take place.
Would you please give me some instructions? Many thanks?
These pictures are posted on a Chinese website, and I hope they can be seen to everyone.