Is it true, that $\mathbb{R} \times \mathbb{Q} \sim \mathbb{R}$? Just like in title, is it true that $\mathbb{R} \times \mathbb{Q} \sim \mathbb{R}$?
My answer would be yes, since $\mathbb{R}^2 \sim \mathbb{R}$.

Without going through $\mathbb R\times\mathbb R$:


*

*Inject $\mathbb R$ into $(0,1)$ with $f(x)=\frac{1}{2}+\frac{1}{\pi}\arctan x$

*Inject $(x,p/q)\in\mathbb R\times\mathbb Q$, with $\gcd(p,q)=1$ and $q>0$ into $2^p(2q+1)+f(x)$ (if $p<0$, then inject it into $-2^{|p|}(2q+1)+f(x)$


So you get an injection $\mathbb R\times\mathbb Q\to\mathbb R$. The other injection is trivial. By Cantor-Bernstein, they have the same cardinality.
