Closed and Compactness on $\mathbb Q$ (Multiple Choice)

Question

Please help me regarding the following question.

Consider $\mathbb Q$ with usual metric (i.e $d(p,q)=|p-q|$).Then which of the following are true?

1. $\{q\in\mathbb Q|2<q^2<3\}$ is closed
2. $\{q\in\mathbb Q|2\le q^2 \le4\}$ is compact
3. $\{q\in\mathbb Q|2\le q^2 \le4\}$ is closed
4. $\{q\in\mathbb Q|q^2 \ge1\}$ is compact

I was confused about the closed and compactness in $\mathbb Q$. Actually can I use "closed and bounded iff compact " here?I mean what is the easy way to show some set is NOT compact?

Thnx.

Answer 1

1. The complement is open (because $\sqrt2,\sqrt 3$ are irrational), hence the set is closed (even though it is also open)

2. The sets $U_n=\{\,q\in\mathbb Q\mid q^2>2+\frac1n\,\}$ are an open cover (because $\sqrt 2\notin \mathbb Q$), but do not allow a finite subcover; hence no

3. Again, the complement is open

4. The sets $U_n=\{\,q\in\mathbb Q\mid q^2<n\,\}$ are an open cover, but do not allow a finite subcover; hence no