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

Let's assume known the fact that $\Bbb R^m$ and $\Bbb R^n$ have the same cardinality for all pairs of positive integers $(m,n)$.
Then the following follows easily: Let $X\subset\Bbb R^n$ such that there exists a subset $A\subset X$ homeomorphic to an open subset of $\Bbb R^m$ for some $m\leq n$. Then $|X|=|\Bbb R^n|$.
Indeed, such an $A$ obviously contains an open segment $I$ (e.g. any radius of any open ball contained in $A$) and the chain of inclusions
$$
I\subset A\subset X\subset\Bbb R^n
$$
together with the fact that $|I|=|\Bbb R|=|\Bbb R^n|$ shows that $|X|=|\Bbb R^n|$.
It is clear that both the ball $B$ and the torus $T$ (no matter whether you take the solid shapes or just their surfaces) satisfy the property required for $X$, thus $|B|=|\Bbb R^3|=|T|$.
