Partition of $\mathbb R$ into uncountably many countably infinite subsets such that their limit points are not members of them

I only scratched some introductions of set theory so this could be a very naive question.

But I would like to know is such a partition possible at all?

That is, does there exist a partition of $$\mathbb R$$ such that:

a) That partition consists of uncountably many $$\{S_i: i \in I\}$$ subsets of $$\mathbb R$$

b) Every $$S_i$$ is countably infinite

c) If $$w$$ is a limit point of $$S_i$$ then $$w \notin S_i$$

Yes. Consider $$S_i:=\{n+i:n\in\mathbb{Z}\}$$ where $$i\in[0,1)$$. a) and b) can be checked easily, and there are no limit points for any $$S_i$$, c) is satisfied.