The one-point compactification $\mathbb{N}_\infty$ of $\mathbb{N}$ is obtained from the discrete space $\mathbb{N}$ by adjoining a limit point $\infty$. It may be identified with the subspace of Cantor space $$ \mathbb{N}_\infty = \{ \alpha \in \{0,1\}^\mathbb{N} \mid \forall n \,.\, \alpha_n \geq \alpha_{n+1} \}. $$ Indeed, we may embed $\mathbb{N} \to \mathbb{N}_\infty$ by mapping $n$ to the sequence $$\overline{n} = \underbrace{1 \cdots 1}_n 0 0 \cdots,$$ and taking $\infty = 1 1 1 \cdots$.

Classically of course adjoining a single point to a countable set has no effect on countability. How about the computable version? If we adjoin the new point as an isolated one then of course we again obtain a countable set. This question is about adjoining $\infty$ as a limit point.

Let $\varphi$ be a standard enumeration of partial computable maps.

Question: Do there exist a total computable map $q$ and a partial computable map $s$ such that:

1. $\varphi_{q(n)} \in \mathbb{N}_\infty$ for all $n \in \mathbb{N}$
2. For all $k \in \mathbb{N}$, if $\varphi_k \in \mathbb{N}_\infty$ then $s(k)$ is defined and $\varphi_{q(s(k))} = \varphi_k$.

The map $q$ realizes an enumeration $\mathbb{N} \to \mathbb{N}_\infty$, and $s$ the fact that $q$ is surjective.