More a verification than a question, really.
The Baire category theorem says that in a Baire space a set of the first category has no interior. (Dugundji, Topology, p. 250.)
Does that mean that: ($A$ is of the first category) $\Longleftrightarrow$ ($int A = \emptyset$)?
No. It means that$$A\text{ is of first category}\color{red}{\implies}\mathring A=\emptyset.$$It is not an equivalence. For instance, the interior of $\mathbb R\setminus\mathbb Q$ is empty (in $\mathbb R$), but $\mathbb R\setminus\mathbb Q$ is not of first category.