TITLE: $(0,1)\cap \mathbb{Q}$ is not compact?
QUESTION [7 upvotes]: I hope you would read my approach.
First, choose an increasing sequence, $\{a_n\}$ of irrational numbers, the limit of which is $1$. Then, the open covering, $\{(0,a_1),(a_1,a_2),(a_2,a_3),\ldots\}$ has no finite subcovering.
Therefore, $(0,1)\cap \mathbb{Q}$ is not compact.

REPLY [5 votes]: I suppose you meant an increasing sequence of irrational numbers. Yes, your approach works. Here's another one: it's not compact because it's not closed (take $x_n = 1/n$).