If one intersects two sets of positive Hausdorff dimension, then the
resulting set is usually of smaller Hausdorff dimension than those of the
I will give an overview of two different types of classes of sets with the
following property: There is a number s such that any set in the class has
Hausdorff dimension at least s, and any countable intersection of sets from the
class belongs to the class.
Some applications in dynamical systems and metric number theory will be