The Continuity Of Some Metric Invariants In Hausdorff And Gromov-Hausdorff Topology

Hausdorff distance and it’s generalization, Gromov-Hausdorff distance are important tools in metric geometry, which have important applications in other mathematical fields. In this paper, we will study the continuity of some metric invariants in Hausdorff distance and Gromov-Hausdorff distance. We first give the detailed proofs of continuity of some simple metric invariants. M. Gromov introduced the notion of the k-width(0< k< n) of a subset in Euclidean space Rn, and prove that the mean (n-1)-width is continuous in Hausdorff distance. It is well known that Lipchitz topology and uniform convergence topology are stronger than Gromov-Hausdorrf topology. In this paper, we prove that in the space of paths of a metric space, path uniform convergence topology is not stronger than Gromov-Hausdorff topology. Finally, we point out a hole in proof of continuity of covering radius in [5].