Can a bijection be constructed between $\mathbb{Q}$ and $\mathbb{R}$, such that $f:\mathbb{Q} \to \mathbb{R}$ is a bijective function?
I understand that there exists no bijection between $\mathbb{N}$ and $\mathbb{R}$, and that the real numbers are not a countable set, however, since the rational numbers form a dense subset of the real numbers, I wondered if some bijective function might exist?
Best Answer
No. One may prove that $\Bbb Q$ is countable while $\Bbb R$ is uncountable. Hence there is no bijection $\Bbb Q\to\Bbb R$.
Topological spaces containing a countably dense subspace are interesting enough that they have earned a name. Spaces with this property are called separable.