Why is $\bar{\mathbb B}^n/\mathbb S^{n-1}$ homeomorphic to $\mathbb S^n$ and how to visualize a quotient map making the same identifications?

Question

I'm currently studying Lee's Introduction to Topological Manifolds. In Chapter - 4 (Example 4.55), he proves that $\bar{\mathbb B}^n/\mathbb S^{n-1}$ is homeomorphic to $\mathbb S^{n}$ using the Closed Map Lemma. I had some confusions regarding his proof.

• Firstly, how to "properly" visualize the space $\bar{\mathbb B}^n/\mathbb S^{n-1}$ ? Why does it not just look like this:
  
  Since we are just collapsing the whole boundary to a single point, why is the above visualization wrong? (The figure refers to the case when $n=2$.)
• Secondly, Lee claims that the map $q:\overline{\mathbb B}^n\to\mathbb S^n$ given by $$ q(x) = \left( 2x\sqrt{1-|x|^2},2|x|^2-1 \right) $$ is a quotient map. To apply the closed map lemma, we first need to show that $q$ defined as such is actually a continuous surjective map. What's the simplest way to quickly/intuitively make this observation? How can someone come up with these maps in the first place?

TIA.

Answer 1

For the first question, imagine the ball (e.g. in dim 2) to be a disc of fabric. On its edge (so the sphere of dim n-1 (here the circle), you sew a drawstring hem, then you pull the cord, which closes the disc on its edge, you then have a closed and empty container homeomorphic to a sphere of dimension 2

For your second question, for $|x| = 1$, you get $q(x) = (0_{n},1) = p_N$ so it is sort of a "inverted" stereographic projection of $\mathbb{S^n} \setminus p_N$ (a pole of the sphere). So if you remove $p_N$ of the sphere, you can invert $q$, then $q^{-1}$ acts like this:

It is like you made a hole at the top of your sphere (which can be of an extensible material if you want) and then enlarged it enough to flatten the whole thing on the ground, which gives you a disc.

For q, it is the inverse transformation, you take your disc (of clay for example) lift its edges, rounding it a bit with your hands progressively until the edge that has become smaller and smaller just reunites and glues to itself.

If you're not satisfied with my explanation, do the math for $n = 1$ on geogebra for example, it should be quite easy to get this afterwards