TITLE: Is the set of rational number discrete or continuous?
QUESTION [10 upvotes]: 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.

REPLY [10 votes]: To say that the reals are continuous does not mean anything. There is no such notion, strictly speaking. Functions are continuous. One could say that the reals are complete. The rationals then, although dense, are not complete.