Show that $\Bbb Q$ is zero-dimensional.
I’m not sure if I’m confusing something, but since $\Bbb Q$ is countable the set $\mathcal{B}=\{\{q\} : q \in \Bbb Q\}$ forms a basis for it? And by definiton a set is zero-dimensional if it has a basis whose elements are closed. So wouldn’t this imply that $\Bbb Q$ is of dimension zero?
No, $\mathcal B$ is not a basis of the usual topology of $\Bbb Q$. Actually, no element of $\mathcal B$ is an open set with respect to that topology.
Instead, take $\mathcal B=\left\{(x,y)\cap\Bbb Q\,\middle|\,x,y\in\Bbb R\setminus\Bbb Q\wedge x<y\right\}$. Then $\mathcal B$ is a basis of the usual topology of $\Bbb Q$. Furthermore, it consists of clopen sets, and therefore $\Bbb Q$ is $0$-dimensional.