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Why are the integers with the cofinite topology not path-connected?

As the title, it is possible to find subsets of $[0,1]$ such that

$\displaystyle [0,1]= \bigcup_{n \in \mathbb N}^{\cdot} C_n $ where $C_n$ are closed and disjoint subset of $[0,1]$ ?

Two observations:

$ \cdot $ if we not consider a *countable* set of subsets it is sufficient to take every point as a subset.

$ \cdot $ if we study the particular case where the subsets are closed intervals the answer is no, Is not possible; to see that is sufficient and not difficult to show that the partition considered is a perfect subset of $\mathbb R$ and so it is not countable. (This particular case is even known as a Sierpiński's Lemma)