A little question on rationals and open sets in $\mathbb{R}$.

Question

Let $A$ be an open set of real numbers, and $D$ be the set of rationals in $A$. For every $d\in D$, let $J(d)$ be an (arbitrary) open interval such that $d \in J(d) \subseteq A$. Is it true that $\bigcup \{J (d) | d \in D\} = A$ ?

Answer 1

No, it isn't true.

Consider $A=(0,1)$ and $J(d)$ being whichever one of $(0,\frac1\pi)$ or $(\frac1\pi,1)$ which contains $d$.