  We show that in a generic, ergodic quantum many-body system the interactions
induce a non-trivial topology for an arbitrarily small non-hermitean component
of the Hamiltonian. This is due to an exponential-in-system-size proliferation
of exceptional points which have the hermitian limit as an accumulation
(hyper-)surface. The nearest-neighbour level repulsion characterizing hermitian
ergodic many-body sytems is thus shown to be a projection of a richer
phenomenology where actually all the exponentially many pairs of eigenvalues
interact. The proliferation and accumulation of exceptional points also implies
an exponential difficulty in isolating a local ergodic quantum many-body system
from a bath, as a robust topological signature remains in the form of
exceptional points arbitrarily close to the hermitian limit.
