The closure of this set in $[0,1]$ is exactly the subset $mathcal \mathcal C$ of all real numbers whose continued fraction expansions involve only different numbers. This last Rational numbers are fine for membership in $\mathcal C$ if they have a continued fraction expansion involves only distinct integers.
The set $\mathcal C$ is a Kantor set . By the way this last set and it is also the closure of all real numbers whose continued fraction is a permutation of $\mathbb N$ with finite support (moving only identity outside a finite number of elements)set).
