Example 4, Sec. 22 in Munkres' TOPOLOGY, 2nd edition: How to figure out saturated open sets? Let $X$ be the closed unit ball 
$$ \{ \ x \times y \ \colon \ x^2 + y^2 \leq 1 \ \} 
$$
in $\mathbb{R}^2$, and let $X^*$ be the partition of $X$ consisting of all the one-point sets $\{ \ x \times y \ \}$ for which $x^2 + y^2 < 1$, along with the set $S^1 = \{ \ x \times y \ \colon \ x^2 + y^2 = 1 \ \}$. Then the map $p \colon X \to X^*$ defined by 
$$
p(x) = 
\begin{cases} 
\{ \ x \times y \ \} \ & \mbox{ if } \ x^2 + y^2 < 1, \\
S^1 \ & \mbox{ if } \ x^2 + y^2 = 1
\end{cases}
$$
is surjective, and the topology of $X^*$ is to be defined so as to make $p$ into a quotient map, that is, so that a subset $U$ of $X^*$ is open in $X^*$ if and only if $p^{-1}[U]$ is open in $X$. 
Now how to characterise the open sets in $X$ that are saturated with respect to $p$? That is, how to decide if a subset $A$ of $X$ is a saturated open set? 
To be frank, I'm not even at facility with the open sets in $X$, although I know that a subset of $X$ is open in $X$ if and only if this subset can be expressed as the intersection of $X$ with an open set in $\mathbb{R}^2$. 
And, how to show that $X^*$ is homeomorphic with the unit 2-sphere in $\mathbb{R}^3$ given by 
$$
\{ \ (x, y, z) \ \colon \ x^2 + y^2 + z^2 = 1 \ \}?
$$
 A: Note that the sets of $X$ that are saturated with respect to $p$ are the sets that either


*

*contain all of $S^1$


or


*

*are disjoint from $S^1$.


Thus, the saturated open sets are the open sets on the interior $\{(x,y): x^2 + y^2 < 1\}$, as well as the open sets containing $S^1$.  An example of a latter such set is $X \cap \{(x,y): x^2 + y^2 > r\}$ for some $0\le r<1$.  
You might try to prove that the images of the latter form map in the quotient to a neighborhood basis for the point given by the equivalence class of $S^1$.  (This relies on the compactness of $S^1$, I think)  Remember that open sets of $X$ are of the form $U \cap X$ for $U$ open in $\Bbb R^2$, so a basis is given by $B \cap X$ where $B$ is some open ball.
You should be able to describe the desired homeomorphism by first demonstrating a homeomorphism between the open ball and the sphere minus a point.  Then you send the equivalence class of $S^1$ to the remaining point, and have to check that this is a homeomorphism.
