  Based on a quantitative version of the classical Hopf-Rinow theorem in terms
of the doubling property, we prove a new precompactness principle in the
(pointed) Gromov-Hausdorff topology for domains in (maybe incomplete)
Riemannian manifolds with a lower Ricci curvature bound, which are applicable
to those with a Reifenberg-flat boundary, or the covering spaces of balls
naturally appear in the study of local geometry and topology of manifolds with
lower curvature bounds.
