  In this paper, we show that, the family of all possible union of finite
consecutive cylinders of the same rank of $Q_{\infty}$-expansion is faithful
for the Hausdorff dimension calculation. Applying this result, we give the
necessary and sufficient condition for the family of all cylinders of
$Q_{\infty}$-expansion to be faithful for Hausdorff dimension calculation on
the unit interval, this answers the open problem mentioned in a paper of S.
Albeverio et al..
