I need to solve the following question:
Find an open cover for $\mathbb{Q} \cap [0,1] \subset (\mathbb{R}, T_{st})$ that doesn't contain a finite subcover, where $T_{st}$ is the standard topology.
Q: How do I solve this?
I've tried to solve this question but I didn't succeed. Help would really be appreciated, thanks in advance!
Best Answer
Pick an irrational $\alpha$ in the unit interval. For each rational $r$ find an open interval $U_r$ that contains $r$ but avoids $\alpha$. Then the collection $\{U_r\}$ form an open cover of the rationals. But any finite subcollection will fail to cover the rationals, since the union of a finite number of intervals $U_r$ will stay a positive distance away from $\alpha$; in that nonzero gap will be at least one rational.