What are the conected components of $\displaystyle Q$ with topology from $\displaystyle \Re$?

I can prove that the inteior of $\displaystyle Q$ in $\displaystyle \Re$ is the empty set. I think that any subset A of $\displaystyle \Re$ such that the interior of A is the empty set has connected components that are te singletons, but cannot prove this. Can anyone verify that this idea is correct please? Thanks.