Convergence of extremal subsets in Alexandrov spaces

Question

Let $\{X_i^n\}$ be a sequence of $n$-dimensional Alexandrov spaces with curvature uniformly bounded from below which converges in the Gromov-Hausdorff sense to a compact $n$-dimensional Alexandrov space (i.e. without collapse). Let $E_i\subset X_i$ be extremal subsets. Assume that $E_i$ converge to a compact subset $E\subset X$ in the Hausdorff sense.

Is it true that $E$ is an extremal subset of $X$?

Answer 1

The limit of extremal subsets is an extremal subset, see Lemma 4.1.3 in Petrunin's Semiconcave functions in Alexandrov geometry. The non-collapsing assumption is not needed.