  A labeled metric space is intuitively speaking a metric space together with a
special set of points to be understood as the geometric boundary of the space.
We study basic properties of a recently introduced labeled Gromov-Hausdorff
distance, an extension of the classical Gromov-Hausdorff distance, which
measures how close two labeled metric spaces are.
  We provide a toolbox of results characterizing convergence in the labeled
Gromov-Hausdorff distance. We obtain a completeness result for the space of
labeled metric spaces and precompactness characterizations for subsets of the
space. The results are applied to travel time inverse problems in a labeled
metric space context.
