Now you seem to be putting a topology on R^2 (Sorgenfrey plane). What is a rational number in R^2? Do you mean elements of QxQ? If so, then the original hint applies and yes, your answer to the question is correct. Can you finish from there?
If every set in a collection contains non-rationals, then any union of those sets must contain non-rationals. Every open set is such a union, and so...?