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.