Metrization of Gromov-Hausdorff-type topologies on boundedly-compact metric spaces

We present a new general framework for metrization of Gromov-Hausdorff-type topologies on non-compact metric spaces. We also give easy-to-check conditions for separability and completeness and hence the measure theoretic requirements are provided to study convergence of random spaces with additional random objects. In particular, our framework enables us to define a metric inducing a suitable Gromov-Hausdorff-type topology on the space of rooted boundedly-compact metric spaces with laws of stochastic processes and/or random fields, which was not clear how to do in previous frameworks. In addition to general theory, this paper includes several examples of Gromov-Hausdorff-type topologies, verifying that classical examples such as the Gromov-Hausdorff topology and the Gromov-Hausdorff-Prohorov topology are contained within our framework.