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Is there a partition of the real numbers into infinitely many closed subsets so that no infinite union of these subsets (other than the whole set of real numbers) is closed?

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    I'd think that if the answer is positive it lies within Cantor set sort of sets.2012-02-27
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    Note that it is impossible to write $\mathbb R$ as the disjoint union of countably many (but $\ge 2$) nonempty closed sets. So you'll need an uncountable partition.2012-02-27
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    Another observation is that almost all the sets needs to be very close to one another at some point, if you had infinitely many sets which are separated by pairwise disjoint open sets then their union would be closed.2012-02-28

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