Vtyjkyuk

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 $X=mathbbQ$ be the set of rational numbers and let $d$ be standard Euclidean metric on $mathbbQ$

A very important result:
Restrict a metric, gives same topology as subspace topology from larger space X

Example 1: $A=rinmathbbQ: r^2<2$

Is $A$ closed?

A is closed in $mathbbQ$ as $A=mathbbQcap [-sqrt2,sqrt2]$

Is $A$ open?

$A$ is open in $mathbbQ$ as $A=mathbbQcap(-sqrt2,sqrt2)$

Is $A$ compact and connected?

I am stuck here. I know the definitions but I am not able to proceed.

Setting $A_s=rinmathbb Qmid r^2<s=mathbb Qcap(-sqrt s,sqrt s)$ it is evident that the sets $A_s$ are open for $0<s<2$ and cover $A$.

Can you find a finite subcover?

We have $A=rinmathbb Qmid r^2<frac12cup rinmathbb Qmid frac12<r^2<2$ since $rinmathbb Qmid r^2=frac12=varnothing$.

The sets are open, not empty and disjoint.

Not compact. Consider the open cover

$mathbbQcap(-sqrt2,r)$: r in A

Not connected. Let r be an irrational in $(-sqrt2,sqrt2)$ and consider
$(-sqrt2,r), (r,sqrt2)$ both intersected by Q.