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 $X$?**



Convergence of extremal subsets in Alexandrov spaces
The limit of extremal subsets is an extremal subset, see Lemma 4.1.3 in Petrunin's [Semiconcave functions in Alexandrov geometry](https://arxiv.org/abs/1304.0292). The non-collapsing assumption is not needed.


