Let $A$ be the set of all rational $p$ such that $2 < p^2 < 3$.Then $A$ is choose the correct option

Question

Let $A$ be the set of all rational $p$ such that $2 < p^2 < 3$.Then $A$ is

choose the correct option

$1)$ compact in $\mathbb{Q}$

$2)$closed and bounded in $\mathbb{Q}$

$3)$Not compact in $\mathbb{Q}$

$4)$ closed and unbounded in $\mathbb{Q}$

i thinks option $1)$ and $2)$ option will be correct by Heine Boral theorem

Is its True ?

Answer 1

The set $A$ is not closed in $\mathbb R$. For instance, $\sqrt2$ belongs to $\overline A$, but not to $A$. So, $A$ is not compact and 1) doesn't hold. Besides, $A$ is clearly bounded and also closed in $\mathbb Q$, so 2) holds and 4) also hold. The correct choices are 2) and 3). If you want to show directly that $A$ is not compact (in $\mathbb Q$ or in $\mathbb R$, it doesn't matter), use the fact that it is not a closed set.