Studying topology on $\mathbb{Q}$ with an example I am studying mathematics on my own and today I am learning topology of rationals.
I want to analyze the topology through some examples. 
Let $X=\mathbb{Q}$ be the set of rational numbers and let $d$ be standard Euclidean metric on $\mathbb{Q}$ 
A very important result:
Restrict a metric, gives same topology as subspace topology from larger space X
Example 1: $A=\{r\in\mathbb{Q}: r^2<2\}$
Is $A$ closed? 
A is closed in $\mathbb{Q}$ as $A=\mathbb{Q}\cap [-\sqrt{2},\sqrt{2}]$
Is $A$ open? 
$A$ is open in $\mathbb{Q}$ as $A=\mathbb{Q}\cap(-\sqrt{2},\sqrt{2})$
Is $A$ compact and connected? 
I am stuck here. I know the definitions but I am not able to proceed. 

Not compact. Consider the open cover
{ $\mathbb{Q}\cap(-\sqrt{2},r)$:  r in A }  
Not connected.  Let r be an irrational in $(-\sqrt{2},\sqrt{2})$ and consider 
$(-\sqrt{2},r), (r,\sqrt{2})$ both intersected by Q.
