Is the set of rational number discrete or continuous? If the set of real numbers $\Bbb{R}$ is continuous, and the set of integer $\Bbb{Z}$ is a discrete set, then is the set of rational number $\Bbb{Q}$ continuous or discrete? My question is stated in the context of analysis. Sorry if I can’t state my problem clear, this question just passed my mind. If it is just a nonsense question, please tell me right away. Thank you very much.

This depends on the topology that we equip $\mathbf{Q}$ with. If it has its usual topology, i.e. the topology inherited from the standard topology on $\mathbf{R}$, then it is not discrete. A topological space $X$ is said to be discrete if given any $x\in X$ there exists an open set $U$ containing $x$ such that $U\cap X=\{x\}$. Given any $\frac{p}{q}\in \mathbf{Q}$, and an open neighborhood of radius $\epsilon$, we can find another rational $\frac{m}{n}$ satisfying $\lvert \frac{p}{q}-\frac{m}{n}\rvert<\epsilon$, so that $\mathbf{Q}$ is not discrete.
