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I am trying to figure out why the metric space $\mathbb{R}$ with the standard metric cannot be written as a countable union of nowhere dense sets.

Then, another natural question is: Can we write $\mathbb{Q}$ as a countable union of nowhere dense sets?

Can anybody help me with this?

Thanks for giving me time.

Edited: I need a little more explanation. It seems that it is a consequence of Baire category theorem, but I haven't studied this before.

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    @JonasMeyer oh i got your point. Many many thanks to you for such a long discussion.2012-05-29

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