Question about working area of Vitali cover

Question

$\text Carothers' Real Analysis$ defines his Vitali cover with no introduction which made me confused a lot.

Here is the definition of a Vitali Cover:

I'm not sure when does a set have its Vitali Cover? It seems any subset of $\mathbb R$ does exist its Vitali Cover. If it is true, how about $E$ be the set of all rationals in $\mathbb R$(or say rational set $\mathbb Q$)? What kind of Vitali Cover does $\mathbb Q$ have?

P.S. Symbol "$m$" denote Lebesgue Measure that is Lebesgue Outer Measure under Lebesgue measurable set.

Answer 1

I'm sure that any subset of $\mathbb R$ does exist its Vitali Cover(Each element of Vitali Cover, the interval, is closed). For each $x ∈ X ⊂ \mathbb R$, let $I^{x}_{m} = [x, x+1/m]$ for $∀$ positive integer $m$ and let $C =$ {$I^{x}_{m}: x∈X$ and $m=1,2,3,...$}. Then $C$ forms a Vitali cover for $X$.