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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 $w \notin S_i$

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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.

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    $\begingroup$ You beat me by 8 seconds. $\endgroup$ Mar 10 '20 at 23:38
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    $\begingroup$ It was a photo-finish with me, but I'll defer to you, An Jin. $\endgroup$
    – Rob Arthan
    Mar 10 '20 at 23:46

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