It's trivial that we can have a subset $\displaystyle S \subset \mathbb{R}$ such that any bounded neighbourhood of $\displaystyle 0$ contains $\displaystyle \aleph_0$ elements of $\displaystyle S$, and such that the complement of any bounded neighbourhood of $\displaystyle 0$ contains finitely many elements of $\displaystyle S$. Now is it possible to replace "$\displaystyle \aleph_0$" by "$\displaystyle \aleph_1$" and "finitely many" by "$\displaystyle \aleph_0$"?