  We discuss topology in dissipative quantum systems from the perspective of
quantum trajectories. The latter emerge in the unraveling of Markovian quantum
master equations and/or in continuous quantum measurements. Ensemble-averaging
quantum trajectories at the occurrence of quantum jumps, i.e., the jump times,
gives rise to a discrete, deterministic evolution which is highly sensitive to
the presence of dark states. We show for a broad family of
translation-invariant collapse models that the set of dark state-inducing
Hamiltonians imposes a nontrivial topological structure on the space of
Hamiltonians, which is also reflected by the corresponding jumptime dynamics.
The topological character of the latter can then be observed, for instance, in
the transport behavior, exposing an infinite hierarchy of topological phase
transitions. We develop our theory for one- and two-dimensional two-band
Hamiltonians and show that the topological behavior is directly manifest for
chiral, PT, or time reversal-symmetric Hamiltonians.
