  The famous Hopf-Rinow Theorem states, amongst others, that a Riemannian
manifold is metrically complete if and only if it is geodesically complete. The
Clifton-Pohl torus fails to be geodesically complete proving that this theorem
cannot be generalized to compact Lorentzian manifolds. On the other hand, Hopf
and Rinow characterized metric completeness also by properness.
Garc\'ia-Heveling and the author recently obtained a Lorentzian
completeness-compactness result for open manifolds with a similar flavor. In
this manuscript, we extend the null distance used in this approach and our
theorem to proper cone structures and to a new class of semi-Riemannian
manifolds, dubbed $(n-\nu,\nu)$-spacetimes. Moreover, we demonstrate that our
result implies, and hence generalizes, the metric part of the Hopf-Rinow
Theorem.
