  We prove that the mapping class group of a connected oriented infinite-type
surface with a non-empty boundary is left-orderable. Our main strategy involves
the inductive construction of an ideal arc system for the surface using a
carefully chosen exhaustion by finite-type subsurfaces. In fact, we prove that
a generalised ideal arc system for the surface also induces a left-ordering on
the big mapping class group. We then prove that two generalised ideal arc
systems determine the same left-ordering if and only if they are loosely
isotopic. Finally, we prove that the topology on the big mapping class group is
the same as the order topology induced by a left-ordering corresponding to an
inductively constructed ideal arc system.
