Set equivalence using Cantor-Bernstein theorem

Question

(a) Prove that the two-dimensional sphere $S^2:x_1^2+x_2^2+x_3^2= 1$ is equivalent to the two-dimensional torus $T^2$.

(b) Prove that the following two sets in $R^3$ are equivalent and both have power of continuum $c$: the open cube $C_o: 0< x <1; 0< y <1;0< z <1$ and the closed ball $B_c:x^2+y^2+z^2 \leq 1$

If we use Cantor-Bernstein theorem, all we need to find are injective maps from $A$ to $B$ and $B$ to $A$ and then we can say that $A$ is equivalent to $B$. I'm not sure how to construct these maps. Do we use parametric equations?

Answer 1

Here's one way:

(a) I'll think about parameters like you suggested. The sphere is given by two parameters, $0 \leq \theta \leq \pi$ and (for $0 < \theta < \pi$) $0 \leq \phi \leq 2\pi$, while the torus is given by two parameters $0 \leq \theta \leq 2\pi$ and $0 \leq \phi \leq 2\pi$. To injectively map the sphere into the torus, map $\theta$ to $\theta$ and $\phi$ to $\phi$ (with a choice of $\phi$ for the poles). To injectively map the torus into the sphere, map $\theta$ to $\frac{\pi+\theta}{4}$ and $\phi$ to $\phi$.

(b) Both maps can be constructed by shrinking and translating.

Answer 2

Both are subsets of $\mathbb{R}^3$, whose cardinality is the same of that as $\mathbb{R}$. Both contain copies of $S^1$ whose cardinality is the same of that as $\mathbb{R}$.