# How much choice is necessary to prove this statement?

Consider the following statement (in $$\mathsf{ZF}+\text{AC}_\omega (\mathbb{R})$$):

There exists $$(\varphi_\alpha)_{\alpha\in\omega_1}$$ with $$\varphi_\alpha : \alpha \rightarrow \mathbb{N}^\mathbb{N}$$ injective and $$\text{ran}(\varphi_\alpha)$$ is closed and $$\text{rank}_{CB}(\text{ran}(\varphi_\alpha)) = 1$$, i.e. $$\text{ran}(\varphi_\alpha)$$ is made of isolated points, for all $$\alpha\in \omega_1$$.

where $$\mathbb{N}^\mathbb{N}$$ is the Baire space (the space of infinite sequences of natural numbers with its usual topology) and $$\text{rank}_{CB}$$ is the Cantor-Bendixon rank. The statement trivially holds if we assume $$\text{AC}_{\omega_1}$$.
My questions are:

1. How much choice do we need (at least) to prove the above statement? Or more generally what known "weak" axiom (on top of $$\mathsf{ZF}+\text{AC}_\omega (\mathbb{R})$$) can be assumed to prove the statement?
2. Is it equivalent to the statement "There exists an $$\omega_1$$ subset of the reals"? In case it is not, is it consistent $$\mathsf{ZF}+\text{AC}_\omega (\mathbb{R})+$$ The above statement $$+$$ "It does not exists an $$\omega_1$$ subset of the reals"?

Thanks!

• By the way, you might be interested to know that existence of an $\omega_1$-sequence of reals is equivalent to the weaker assertion that that there is a choice of injections $(\varphi_{\alpha})_{\alpha<\omega_1}: \alpha \rightarrow \mathbb{R}.$ Nov 14, 2021 at 1:10

Your statement is equivalent to the assertion that there is a function choosing an enumeration of every countable ordinal. From an enumeration of $$\alpha,$$ you can easily inject it into a countable set of isolated points in Baire space. For the other direction, if $$\varphi_{\alpha}$$ injects $$\alpha$$ into Baire space, with the range a set of isolated points, there is a surjective partial map from basic open sets onto $$\alpha,$$ by sending $$U \mapsto \beta$$ if $$\beta$$ is unique such that $$\varphi_{\alpha}(\beta) \in U.$$ From this we can easily enumerate $$\alpha.$$ Note that by the Cantor-Bendixson analysis, any closed countable set of reals can be canonically enumerated, by sending basic open sets to the unique point in $$U$$ of maximal rank if there is such a point.
This principle implies there is an $$\omega_1$$-sequence of reals, since an enumeration of an ordinal can be canonically coded by a real. It is strictly stronger than the existence of an $$\omega_1$$-sequence of reals. For example, it clearly implies $$\omega_1$$ is regular, while in Figura's model ($$\mathcal{M}36$$ in Consequences of the Axiom of Choice), there is an $$\omega_1$$-sequence of reals yet $$\omega_1$$ is singular.
Note that everything above is purely in ZF. The possibility remains that countable choice for reals plus an $$\omega_1$$-sequence of reals implies your principle, though that seems unlikely. I would guess adding $$\omega_1$$ Cohen reals to the Solovay model would result in a model with an $$\omega_1$$-sequence of reals and DC but no choice of enumerations for every countable ordinal. I have not verified this however.