The closure of this set in $[0,1]$ is exactly the subset $mathcal C$ of all real numbers whose continued fraction expansions involve only different numbers. This last set is a Kantor set. By the way this last set is also the closure of all real numbers whose continued fraction is a permutation of $\mathbb N$ with finite support (moving only a finite number of elements).

The closure of this set in $[0,1]$ is exactly the subset $\mathcal C$ of all real numbers whose continued fraction expansions involve only different numbers. 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 and it is also the closure of all real numbers whose continued fraction is a permutation of $\mathbb N$ with finite support (identity outside a finite set).

The closure of this set is exactly the set of all real numbers whose continued fraction expansions involve only different numbers. This last set is a Kantor set. By the way this last set is also the closure of all real numbers whose continued fraction is a permutation of $\mathbb N$ with finite support (moving only a finite number of elements).

The closure of this set in $[0,1]$ is exactly the subset $mathcal C$ of all real numbers whose continued fraction expansions involve only different numbers. This last set is a Kantor set. By the way this last set is also the closure of all real numbers whose continued fraction is a permutation of $\mathbb N$ with finite support (moving only a finite number of elements).