Uniform Boundedness on Extremal Subsets in Alexandrov Spaces

In this paper, we study extremal subsets in Alexandrov spaces with dimension n, curvature \(\ge \kappa \) $\ge \kappa$ , and diameter \(\le D\) $\le D$ . We show that the following three quantities are uniformly bounded above in terms of n, \(\kappa \) $\kappa$ , and D: (1) the number of extremal subsets in an Alexandrov space; (2) the Betti numbers of an extremal subset; (3) the volume of an extremal subset. The proof is an application of essential coverings introduced by Yamaguchi.