A generalization of the
Hausdorff dimension of sets is given by restricting the lengths of the intervals in the
covering family. The dependence of this dimension on the choice of covering family is
studied by considering the set of points in the countable unit cube Iω whose
coordinates are the values of the dimensions of some set for a fixed, countable
collection of covering families. General conditions are given in order that two families
yield the same dimension on each set, and that a covering family give the ordinary
Hausdorff dimension.
