$m$ is the Lebesgue measure. I was thinking that:
$\Bbb Q$ is dense and $m(\Bbb Q)=0<1$ but it fails to be open, but maybe I could construct an open set from this fact, or also using that $(0,1)$ is a dense and open subset but $m(0,1)=1$.
Any hints or ideas will be very appreciated. Thanks.
Best Answer
Enumerate $D=\mathbb Q\cap(0,1)$. For every $n$, consider an open interval of length $2^{-n}\varepsilon$ around the $n$th element of $D$. Call $U$ the union of these intervals. Then: