There are bijections $$ \text{Map}(\mathbb{N},\mathbb{N}) \xrightarrow{\alpha} \text{Map}(\mathbb{N},\mathbb{N})\setminus\{0\} \xrightarrow{\beta} \text{SInc}(\mathbb{N},\mathbb{N}) \setminus\{\text{id}\} \xrightarrow{\gamma} \mathcal{P}_\infty(\mathbb{N})\setminus\{\mathbb{N}\} \xrightarrow{\delta} (0,1) \xrightarrow{\epsilon} \mathbb{R} $$ as follows.
- $\alpha(u)=u$ unless $u$ is constant, in which case $\alpha(u)=u+1$.
- $\text{SInc}(\mathbb{N},\mathbb{N})$ is the set of strictly increasing maps from $\mathbb{N}$ to itself, and $\beta(u)(n)=n+\sum_{i\leq n}u(i)$.
- $\mathcal{P}_\infty(\mathbb{N})$ is the set of infinite subsets of $\mathbb{N}$, and $\gamma(v)=v(\mathbb{N})$.
- $\delta(S)=\sum_{i\in S}2^{-i-1}$.
- $\epsilon(x)=(x-\frac{1}{2})/\sqrt{x(1-x)}$.
We can also give a bijection from $\text{Map}(\mathbb{N},\mathbb{N})$ to the full set $\mathcal{P}(\mathbb{N})$ of all subsets of $\mathbb{N}$, as follows. We first note that the rules discussed above also give bijections $$ \text{Map}(\mathbb{N},\mathbb{N}) \xrightarrow{\beta} \text{SInc}(\mathbb{N},\mathbb{N}) \xrightarrow{\gamma} \mathcal{P}_\infty(\mathbb{N}). $$ We also have a bijection $\zeta$ from the set $\mathcal{P}_0(\mathbb{N})$ of finite subsets of $\mathbb{N}$ to $\mathbb{N}$ itself given by $\zeta(S)=\sum_{i\in S}2^i$. Now for $u\in\text{Map}(\mathbb{N},\mathbb{N})$ we define $\eta(u)\in \mathcal{P}(\mathbb{N})=\mathcal{P}_0(\mathbb{N})\amalg\mathcal{P}_\infty(\mathbb{N})$ by $$ \eta(u) = \begin{cases} \zeta^{-1}(n) & \text{ if } u \text{ is constant with value } 2n \\ \gamma\beta(n) & \text{ if } u \text{ is constant with value } 2n+1 \\ \gamma\beta(u) & \text{ if } u \text{ is not constant. } \end{cases} $$ This gives a bijection $\eta\colon\text{Map}(\mathbb{N},\mathbb{N})\to \mathcal{P}(\mathbb{N})$.