Is the family of co-dense subsets of a space ideal?

Question

A set $A$ in a space $(X,\tau)$ is called co-dense if $int(A)=\emptyset$. In this paper, it is mentioned the statement "ideal of co-dense". I am asking, how the family of co-dense subsets of a space is ideal. As the co-dense sets are not closed under finite union. Take rationals and irrational in $\mathbb{R}$ with natural topology, both of them are co-dense but their union is not.

Answer 1

The paper asks for an ideal of codense sets i.e. any ideal consisting of codense sets - but not the family of all codense sets.

What your argument shows is simply that rationals and irrationals cannot lie in the same ideal. This makes sense as they are complementary sets.