# prove that the set of rational numbers is not connected on the real line

Could someone help me through this problem? Prove that the set of rational numbers is not connected on the real line

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Consider the set of rational numbers $q < \sqrt{2}$. Why is it open? Why is it closed? –  Pedro Apr 23 '12 at 3:00
Something even stronger is true: The rationals (with the usual interval topology) are totally disconnected. That is, the only connected subsets of the rationals are the singletons. –  Austin Mohr Apr 23 '12 at 3:04
Consider two separate one with q> root of 2 and the other with q <2 for q root of the real and their union is the set of rational –  Breton Apr 23 '12 at 3:05
$(-\infty,\pi),(\pi,\infty)$ are two open disjoint sets which cover $\mathbb{Q}$.
The same statement could be made for any two intervals of the form $(-\infty, \alpha), (\alpha, \infty)$, where $\alpha\in\mathbb{R}$ is irrational. That being said, I like the elegance of your having chosen a transcendental number. –  Nicholas Stull Apr 23 '12 at 5:17