Quotient topology by identifying the boundary of a circle as one point The following is an example taken from Munkres topology book:

I don't understand why does $X^{*}$is homeomorphic to $S^{2}$, is
this a basic fact that I don't understand or is it an example of something
more advanced ? 
Also, I don't understand figure 22.4, I think that a saturated set
is either contained in $\{(x,y)\mid x^{2}+y^{2}<1\}$ or contains
$S^{1}$and I don't see how this is described in the image, can someone please explain that ?
 A: Imagine that you place a disk on top of a sphere so that the center of the disk is at the north pole. Then spread the disk over the surface of the sphere, and glue the boundary of the disk at the south pole. This is one way to see the homeomorphism that the book is talking about.
To construct such a homeomorphism explicitly, we can use polar coordinates $(r, \theta)$ in $\mathbb R^2$ and cylindrical coordinates $(r, \theta, z)$ in $\mathbb R^3$. Consider the continuous map $f : D^2 \to S^2$ given by
$$
(r, \theta) \mapsto \left(\sqrt{1 - (1 - 2r)^2}, \theta, 1 - 2r\right).
$$
Since the boundary of $D^2$ (corresponding to $r = 1$) is mapped to a single point (the south pole), it follows that $f$ factors through a map $\overline f : X^* \to S^2$, where $X^*$ is $D^2$ with the boundary identified to a single point. By the familiar properties of quotient maps, we see that $\overline f$ is a continuous bijection from a compact space to a Hausdorff space. We conclude that it is the desired homeomorphism.
