Exercise 7.6.3

Enumerate the elements of any countable set $A$ as $\{a_1,a_2,\cdots \}$. Given $𝜖>0$, define the collection of open intervals $O_n$ with $O_i=(a_i-\frac{𝜖}{2^{i+1}},a_i+\frac{𝜖}{2^{i+1}})$.