TITLE: Open cover rationals proper subset of R?
QUESTION [9 upvotes]: If I were to cover each rational number by a non-empty open interval, would their union always be R? It seems correct to me intuitively, but I am quite certain it is wrong. Thanks

REPLY [14 votes]: Maybe this is silly, but for a rational $q<\sqrt 2$ take the open interval to be $(-\infty,\sqrt 2)$, and for a rational $q>\sqrt 2$ take the open interval to be $(\sqrt 2,\infty)$. This would furnish a counterexample.