Is it possible to decompose $\mathbf R$, or in general, any uncountable Polish space, into $\mathfrak c$-many disjoint closed subsets such that the union of any infinite subfamily is dense?
If it is, what additional constraints (if any) have to be put on the space to allow this (I suspect connectedness/zero dimensionality might play an important role there, if it is possible at all)?
I tried to construct such a decomposition using the Bernstein set (or a family of those), but without much success.
This question is motivated by another question (which, as far as I can see, is equivalent to my question in case of $\mathbf R^2$).
edit: As for the last remark, it seems that I was under the false impression that for quotient maps, the closure of preimage is the preimage of closure, which is not true in general.