A definition for `bounded Lorentzian metric space’ is presented and discussed. This is an abstract notion of Lorentzian metric space that is sufficiently general to comprise compact causally convex subsets of globally hyperbolic (smooth) spacetimes, and causets.

It is shown that a generalization of the Gromov-Hausdorff distance and convergence can be applied to these spaces. Furthermore, two additional axioms of timelike connectedness and existence of maximizers, which are stable under GH-convergence, lead to suitable notions of Lorentzian pre-length and length spaces. Similarly, sectional curvature bounds stable under GH-convergence can be introduced. A (pre)compactness theorem is also mentioned and its limitations are discussed. Talk based on joint work with Stefan Suhr (Bochum).