# what does 'arbitrary' mean?

Can arbitrary union of open intervals be written as countable union of open intervals?

Here, I don't even understand what do they mean by arbitrary.

In this case arbitrary just means that there is no restriction on the number or kind of open intervals used to form the union. If you allow all possible unions of open intervals, you get precisely the open subsets of $\Bbb R$. The question asks whether you ever need uncountably many open intervals to form some open set in $\Bbb R$, or whether countably many are always sufficient.
HINT: Consider try using just the countable collection $\mathscr{B}=\{(p,q):p,q\in\Bbb Q\text{ and }p<q\}$ of open intervals with rational endpoints. Is every open interval, no matter what its endpoints, a union of members of $\mathscr{B}$?