  We investigate how topological Chern numbers can be defined when
single-particle states hybridize with continua. We do so exemplarily in a
bosonic Haldane model at zero temperature with an additional on-site decay of
one boson into two and the conjugate fusion of two bosons into one. Restricting
the Hilbert space to two bosons at maximum, the exact self-energy is
accessible. We use the bilinear Hamiltonian $H_0$ corrected by the self-energy
$\Sigma$ to compute Chern numbers by two different approaches. The results are
gauged against a full many-body calculation in the Hilbert space where
possible. We establish numerically and analytically that the effective
Hamiltonian $H_\text{eff}=H_0(\vec k) +\Sigma(\omega,\vec k)$ reproduces the
correct many-body topology if the considered band does not overlap with the
continuum. In case of overlaps, one can extend the definition of the Chern
number to the non-Hermitian $H_\text{eff}$ and there is evidence that the Chern
number changes at exceptional points. But the bulk-boundary correspondence
appears to be no longer valid and edge modes delocalize.
