It is well known that $\mathbb{Q}$ is not a $G_\delta$. In fact no countable dense subset of $\mathbb{R}$ is a $G_\delta$. We order the rationals in a sequence: $\mathbb{Q}=\lbrace r_k\rbrace$, and for each integer $n\ge1$ define $$O_n=\bigcup_{k=1}^\infty(r_k-2^{-(k+n)},r_k+2^{-(k+n)}).$$ $O_n$ is open with measure less than $2^{-n+1}$. Then $$O=\bigcap_{n=1}^\infty O_n$$ is a $G_\delta$ of measure $0$ containig the rationals. We know that $O$ must be uncountable, so that there are uncountably many irrationals in $O\setminus\mathbb{Q}$.

Now the questions. What can we say about any $x\in O\setminus\mathbb{Q}$, apart from the fact that $x$ is irrational? Is it possible to say something about how well it is approximated by rationals? What is the dependence of $O$ on the chosen ordering of $\mathbb{Q}$?

oneof the sets $O_n$... $\endgroup$ – Thierry Zell Feb 6 '11 at 2:37