= Solution
The <Baire category theorem> says that in a nonempty complete <metric space>, a <countable> intersection of open dense subsets is dense. Equivalently, such a space cannot be a <countable> union of <nowhere dense sets>: a set is <nowhere dense> when the interior of its closure is empty. Indeed the complements of those closures are open dense; their intersection is dense and in particular nonempty, so a <countable> union of the original sets cannot exhaust the space. The same argument shows that the complement of such a union is dense.
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