I don't know if it leads anywhere, but you could see if is a closed set or not.
If it's not, then it cannot be nor , which are both open and closed.
Alright this question is from Carothers Real Analysis.
Let be an enumeration of . For each each n, let be the open interval centered at of radius , and let . Prove that U is a proper, open subset, dense subset of and that is nowhere dense in .
Where I am stuck is try to show that is a proper subset of . My first thought was a proof by contradiction, by assuming that . Then by the Baire Categroy theorem one of the but this didn't seem to go anywhere.
Thanks for any input