How to prove that the Torus and a Ball have the same Cardinality How to prove that the Torus and a Ball have the same Cardinality ?
The proof is using the Cantor Berenstein Theorem.
I now that they are subsets of $\mathbb{R}^{3}$ so I can write $\leq \aleph$ but I do not know how to prove $\geqslant  \aleph$.
Thanks

I'm assuming that you're talking about solid balls and tori in $\mathbb R^3$.
To prove that these sets are equipollent using the Cantor–Bernstein–Schröder theorem, all you need to do is to show that:


*

*there exists an injection $f$ from the ball $B$ into the torus $T$, and

*there exists an injection $g$ from the torus $T$ into the ball $B$.


The CBS theorem then says that there exists a bijection between $B$ and $T$.
If you're given explicit geometric definitions of $B$ and $T$, you can use these directly to construct invertible affine maps $f$ and $g$ such that $f(B) \subset T$ and $g(T) \subset B$.
Another, more general way to do this is to note that both of these subsets of $\mathbb R^3$ are bounded and have non-empty interior.  Thus, for each of them, there exists open balls $I$ and $O$ such that $I_B \subset B \subset O_B$ and $I_T \subset T \subset O_T$.  Then choosing $f$ and $g$ (again as affine maps) such that $f(O_B) = I_T$ and $g(O_T) = I_B$ will do the trick.
