Consider a set $X\subseteq \mathbb{R}$ such that

- $X$ is
*not*separable wrt its subspace topology - For all $r\in\mathbb{R}$ there exists a sequence $(x_n)_{n\in\omega} \subset X$ converging to $r$

In a model containing such a set $\text{AC}_\omega(X)$ (choice for countable families of non-empty subsets of $X$) would of course fail, but not that critically.

For example, the unique way I've seen to prove the consistency of the existence of a non-separable set of reals is the one that shows the consistency of an infinite, Dedekind-finite set of reals, but our set, though being non-separable, is well-behaved enough to witness density in a sequencial manner.

My questions are:

- Is its existence consistent relative to $\text{ZF}$? Has it been proved somehwere?
- In case the answer above is "no", does this remind you similar results (besides the most known ones that can be found in Jech'
*Axiom of Choice*)?

Thanks!

thatresult. As an attempt, I'd try adding Cohen reals indexed by $\Bbb Q$, take order preserving automorphisms, and generate the filter by fixing pointwise a sequence, or a bounded sequence. The resulting set of reals is going to be dense insomeinterval at least, so we can stretch it to get a set dense in the reals. So it is enough to show that in its dense area it satisfies (2), which should be doable, I think. Finally, to show it's not separable, I'm not sure if the usual trick is going to work, but it seems like there might be a way to solve that. $\endgroup$