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I have several books that state that a countable intersection of [itex]F_\sigma[/itex] sets need not be an [itex]F_\sigma[/itex] set (indeed, such sets have their own designation, [itex]F_{\sigma\delta}[/itex]), but none of them gives a counterexample. Does anyone know one offhand? This isn't homework/coursework; I'm just curious and haven't been able to come up with one. I would prefer a concrete example if possible, not just an existence proof.

Relevant facts: Clearly a countable UNION of [itex]F_\sigma[/itex] sets is [itex]F_\sigma[/itex]. All open and closed sets in [itex]\mathbb{R}[/itex] are [itex]F_\sigma[/itex]. [itex]\mathbb{Q}[/itex] is an [itex]F_\sigma[/itex] set as it is a countable union of singletons. The only concrete example of a set I know is not [itex]F_\sigma[/itex] is [itex]\mathbb{R}\setminus\mathbb{Q}[/itex], the set of irrationals.