  We present an alternative approach to studying topology in open quantum
systems, relying directly on Green's functions and avoiding the need to
construct an effective non-Hermitian Hamiltonian. We define an energy-dependent
Chern number based on the eigenstates of the inverse Green's function matrix of
the system which contains, within the self-energy, all the information about
the influence of the environment, interactions, gain or losses. We explicitly
calculate this topological invariant for a system consisting of a single 2D
Dirac cone and find that it is half-integer quantized when certain assumptions
over the damping are made. Away from these conditions, which cannot or are not
usually considered within the formalism of non-Hermitian Hamiltonians, we find
that such a quantization is usually lost and the Chern number vanishes, and
that in special cases, it can change to integer quantization.
