  Magnitude homology of graphs is introduced by Hepworth and Willerton in
arXiv:1505.04125 . Magnitude homology of arbitrary metric spaces by Leinster
and Shulman in arXiv:1711.00802v2 . We verify that the K\"unneth and
Mayer-Vietoris formulas proved in arXiv:1505.04125 for graphs extend naturally
to the metric setting. The same is done for the notion of diagonality, also
originating from arXiv:1505.04125 . Stability of this notion under products,
retracts, filtrations is verified, and as an application, it is shown that
median spaces are diagonal; in particular, any Menger convex median space has
vanishing magnitude homology. Finally, we argue for a definition of magnitude
homology in the context of "betweenness spaces" and develop some of its
properties.
