  The effect of a magnetic field on a two-dimensional Chern band insulator is
discussed. It is shown that, unlike the trivial insulator, an anomalous Hall
insulator with Chern number $C$ becomes a metal when a magnetic field is
applied at constant particle density, for any $C>0$. For a time reversal
invariant topological insulator with a spin Chern resolved number, $C_\uparrow=
- C_\downarrow=C$, the magnetic field induces a spin polarized spin Hall
insulator. We consider also the effect of a superlattice potential and extend
previous results for the quantization of the Hall conductance of filled
Hofstadter bands to this problem.
