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?

  • $\begingroup$ No, you can't. Google "Cardinality of sets" $\endgroup$ – DonAntonio May 29 '14 at 19:27
  • $\begingroup$ Here is the prove of the fact that $\mathbb{R}$ is uncountable $\endgroup$ – Math137 May 29 '14 at 19:28
  • $\begingroup$ Please search the site before posting. $\endgroup$ – Asaf Karagila May 29 '14 at 19:38

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.


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