Cylinder sets and continued fractions exhibit the homeomorphism ${\mathbb{N}}^{\mathbb{N}}\cong \mathbb{R}\setminus \mathbb{Q}$

Given: The objects, hypotheses, and choice principles stated above.

Baire sequence space ${\mathbb{N}}^{\mathbb{N}}$ is homeomorphic to the irrational subspace $\mathbb{R}\setminus \mathbb{Q}$. (Baire sequence space is homeomorphic to the irrational real numbers).

For simple continued fractions with $a_0\in \mathbb{Z}$ and $a_n\ge 1$ for $n\ge 1$, the convergents satisfy $p_n=a_n{p}_{n-1}+p_{n-2}$ and $q_n=a_n{q}_{n-1}+q_{n-2}$, with $p_n{q}_{n-1}-p_{n-1}{q}_n=(-1{)}^{n-1}$. A code cylinder $C(a_0,\dots ,a_n)\subseteq {\mathbb{N}}^{\mathbb{N}}$ and the real interval $J(a_0,\dots ,a_n)$ with endpoints $p_n\mathrm{/}{q}_n$ and $(p_n+p_{n-1})\mathrm{/}(q_n+q_{n-1})$ are different objects and are not identified; it is the intervals $J$ that are nested as the prefix is extended, with $\mathrm{diam}J(a_0,\dots ,a_n)=1\mathrm{/}(q_n(q_n+q_{n-1}))\to 0$. (Continued-fraction convergents, determinant identities, and nested irrational cylinders).

The continued-fraction coding of def-simple-continued-fraction-coding is a bijection from the sequences $(a_0,a_1,\dots )$ with $a_0\in \mathbb{Z}$ and $a_n\ge 1$ for $n\ge 1$ onto $\mathbb{R}\setminus \mathbb{Q}$, and both the coding map and its inverse are continuous for the cylinder and subspace topologies (Infinite simple continued fractions parametrise the irrational real numbers).