  Here, we develop a gauge-independent Green function approach to characterize
the Chern invariants of generic non-Hermitian systems. It is shown that
analogous to the Hermitian case, the Chern number can be expressed as an
integral of the system Green function over a line parallel to the
imaginary-frequency axis. The approach introduces in a natural way the
"band-gaps" of non-Hermitian systems as the strips of the complex-frequency
plane wherein the system Green function is analytical. We apply the developed
theory to nonreciprocal electromagnetic continua, showing that the topological
properties of gyrotropic materials are strongly robust to the effect of
material loss. Furthermore, it is proven that the spectrum of a topological
material cavity terminated with opaque-type walls must be gapless. This result
suggests that the bulk-edge correspondence remains valid for a class of
non-Hermitian systems.
