  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.
