An example of quotient space

Question

(Munkres section 22)Example4- Let $X$ be the closed unit ball $$\{x×y|x^2+y^2\leq 1\}$$ in $\Bbb{R}^2$, and let $X^*$ be the partition of $X$ consisting of all the one-point sets $\{x×y\}$ for which $x^2+y^2< 1$, along with the set $S^1=\{x×y|x^2+y^2=1\}$. 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 $\Bbb{R}^3$ called the $unit ~2-sphere$, defined by

$$S^2=\{(x,y,z)|x^2+y^2+z^2=1\}$$

The problem is that how to show that it is saturated?

How $X^*$ is homeomorphic with the subspace of $\Bbb{R}^3$ as given above? I mean how to define a function? Any help.

Thanks!

Answer 1

HINT: Design your homeomorphism $h$ so that it takes:

• $\{\langle 0,0\rangle\}$ to the ‘south pole’, $\langle 0,0,-1\rangle$;
• $\big\{\{\langle x,y\rangle\}:x^2+y^2=r^2\big\}$ to a circle of constant latitude $z=c$ if $0<r<1$, where $c$ ranges over $(-1,1)$ as $r$ ranges over $(0,1)$; and
• $S^1$ to the ‘north pole’, $\langle 0,0,1\rangle$.

You may find it helpful to think in terms of cylindrical coordinates.