Studying topology on $\mathbb{Q}$ with an example

Question

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.

Answer 1

Setting $A_s=\{r\in\mathbb Q\mid r^2<s\}=\mathbb Q\cap(-\sqrt s,\sqrt s)$ it is evident that the sets $A_s$ are open for $0<s<2$ and cover $A$.

Can you find a finite subcover?

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We have $A=\{r\in\mathbb Q\mid r^2<\frac{1}{2}\}\cup \{r\in\mathbb Q\mid \frac{1}{2}<r^2<2\}$ since $\{r\in\mathbb Q\mid r^2=\frac12\}=\varnothing$.

The sets are open, not empty and disjoint.

Answer 2

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.