Possible Duplicate:

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

As in the title, is it possible to find closed, disjoint subsets $C_n$ of $[0,1]$ such that $[0,1] = \bigcup_{n \in \mathbb N} C_n$?

Two observations:

If we do not demand a

*countable*set of subsets, it is sufficient to take every point as a subset.If we consider the particular case where the subsets are closed intervals, then it is not possible. To see that, it 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 SierpiĆski's lemma.