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.

 1. $\alpha(u)=u$ unless $u$ is constant, in which case $\alpha(u)=u+1$.
 2. $\text{SInc}(\mathbb{N},\mathbb{N})$ is the set of strictly increasing maps from $\mathbb{N}$ to itself, and $\beta(u)(n)=n+\sum_{i\leq n}u(i)$.
 3. $\mathcal{P}_\infty(\mathbb{N})$ is the set of infinite subsets of $\mathbb{N}$, and $\gamma(v)=v(\mathbb{N})$.
 4. $\delta(S)=\sum_{i\in S}2^{-i-1}$.
 5. $\epsilon(x)=(x-\frac{1}{2})/\sqrt{x(1-x)}$.

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 $\eta\colon\text{Map}(\mathbb{N},\mathbb{N})\to \mathcal{P}(\mathbb{N})$.