On -Metric Spaces and the -Gromov-Hausdorff Distance

For each given we investigate certain sub-family of the collection of all compact metric spaces which are characterized by the satisfaction of a strengthened form of the triangle inequality which encompasses, for example, the strong triangle inequality satisfied by ultrametric spaces. We identify a one parameter family of Gromov-Hausdorff like distances on and study geometric and topological properties of these distances as well as the stability of certain canonical projections . For the collection of all compact ultrametric spaces, which corresponds to the case of the family , we explore a one parameter family of interleaving-type distances and reveal their relationship with .