Example 4, Sec. 22 in Munkres' TOPOLOGY, 2nd ed: How is this quotient space homeomorphic with $S^2$?

Here is Example 4, Sec. 22, in the book Topology by James R. Munkres, 2nd edition:

Let $X$ be the closed unit ball $$\left\{ \ x \times y \ \vert \ x^2 + y^2 \leq 1 \ \right\}$$ 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 $S^1 = \left\{ \ x\times y \ \vert \ x^2 + y^2 = 1 \ \right\}$. Typical saturated open sets in $X$ are pictured by the shaded regions in Figure 22.4. One can show that $X^*$ is homeomorphic with the subspace of $\mathbb{R}^3$ called the unit-$2$-sphere, defined by $$S^2 = \left\{ \ (x, y, z) \ \vert \ x^2+y^2+z^2 = 1 \ \right\}.$$

Now my question is, how is the space $X^*$ homeomorphic with the subspace $S^2$ of $\mathbb{R}^3$?

I'm sorry but I have not been able to make much sense of Figure 22.4 either; so can someone please also elaborate this to me?

Here are some relevant definitions.

Quotient Map:

Let $X$ and $Y$ be topological spaces; let $p \colon X \to Y$ be a surjective map. The map $p$ is said to be a quotient map provided a subset $U$ of $Y$ is open in $Y$ if and only if $p^{-1}(U)$ is open in $X$. . . .

Saturated Sets:

We say that a subset $C$ of $X$ is saturated (with respect to the surjective map $p \colon X \to Y$) if $C$ contains every set $p^{-1} \left( \ \{ \ y \ \} \ \right)$ that it intersects. Thus $C$ is saturated if it equals the complete inverse image of a subset of $Y$. . . .

Quotient Topology:

If $X$ is a [topological] space and $A$ is a set and if $p \colon X \to A$ is a surjective map, then there exists exactly one topology $\mathscr{T}$ on $A$ relative to which $p$ is a quotient map; it is called the quotient topology induced by $p$.

The topology $\mathscr{T}$ is of course defined by letting it consist of those subsets $U$ of $A$ such that $p^{-1}(U)$ is open in $X$. . . .

Quotient Space:

Let $X$ be a topological space, and let $X^*$ be a partition of $X$ into disjoint subsets whose union is $X$. Let $p \colon X \to X^*$ be the surjective map that carries each point of $X$ to the element of $X^*$ containing it. In the quotient topology induced by $p$, the space $X^*$ is called a quotient space of $X$.

Finally, here is the image of Page 139 of Munkres. I would appreciate a detailed answer rather than a mere hint.

• Informally, you're "collapsing" the circle $S^1$ to one point while keeping the other points in the closed unit ball distinct. Let's say that $S^1$ goes to the "north pole" on the sphere. Then a neighborhood of $S^1$ in the closed unit ball (as depicted by $U$ for example) ends up as a neighborhood of the north pole, while $V$, being closer to the center of the unit ball, ends up nearer to the south pole. This isn't explicit or a proof by any means but more of an intuitive explanation. – m_squared Oct 16 '17 at 13:52
• Imagine wrapping a circle piece of paper around a ball. The boundary of the circle will be "scrunched" at the north pole of the ball. Identifying the points of the boundary of the circle deals with the "scrunching" – Robert Thingum Oct 16 '17 at 13:53