Sets of second category-topology A set is of first category if it is the union of nowhere dense sets and otherwise it is of second category.
How can we prove that irrational numbers are of second category and the rationals are of of first category?

Recall that $\mathbb{R\setminus Q}=\bigcap_{q\in\mathbb Q}\mathbb R\setminus\{q\}$.
This is a countable intersection of dense open sets, and by Baire's category theorem the result is dense, i.e. second-category.
