TITLE: Example for an open set in $\mathbb R$
QUESTION [5 upvotes]: What would be your example of an open proper subset $S \subsetneq \mathbb R$ such that $\mathbb{Q} \subsetneq S$?

REPLY [6 votes]: a) $A_k:=\bigcup_{j=1}^{+\infty}(r_j-2^{-j-k},r_j+2^{-j-k})$ where $\{r_j\}$ is an enumeration of rationals. This show that such that set can be taken as small (in measure) as we want.
b) Take $O_n:=(-1/n,1-1/n)$; then $\{O_n,n\in\Bbb N^*\}$ is an open cover of $[0,1)$ which doesn't have any finite subcover.