I have the following claim:
If $A$ is a closed set that contains every number rational $r$ in $[0,1]$, show that $[0,1]\subset A$.
The prove is easy, Since the set A is closed, all its points are accumulation points, we have to show that every point in $x\in [0,1]$ is an accumulation point in $A$, which I already did.
My question is whether I can weaken the hypothesis, and whether the proposition remains true. I've thought about changing the assumption that set $A$ is open, but I'm not sure if that's true. Any comment is welcome, or if it is possible to deepen the proposition further.