On some metrics compatible with the Fell–Matheron topology

Modifying the Hausdorff–Buseman metric, we obtain a compatible metric with the Fell–Matheron topology on the space of closed subsets of a locally compact Hausdorff second countable space. We also give an alternative expression of this metric, and two more compatible metrics, a metric of summation form and a modified Rockafellar–Wets metric. Through the study of the relation between the original Hausdorff–Buseman metric and the Rockafellar–Wets metric, it is also shown that the convergence in the original Hausdorff–Buseman metric in the space of non-empty closed subsets of RN is equivalent to the Painlevé–Kuratowski convergence.