  This article suggests the definition of 'Lorentzian space' weakening the
notion of Lorentzian length space just as much that it allows for a functor
from the category of causally continuous Lorentzian manifolds to the
corresponding category of Lorentzian spaces, and considers three problems in
the context of maximal Cauchy developments of Lorentzian length spaces (LLSs):
The first is to define pointed Gromov-Hausdorff metrics for spatially and
temporally noncompact LLSs, the second to present an explicit non-spacetime
example of a maximal vacuum Cauchy development in the LLS category, the third
to define canonical representatives for developments. A certain regularity
property for geodesics plays a key role in each of the problems.
