  Considering coupling to a micro-structured bath as a relaxation mechanism in
a periodically driven dissipative Haldane model, we establish that the system
may be tuned to a stroboscopic topological steady state at all finite
temperatures. The amplitude and frequency of the periodic drive is so chosen
that the Floquet Hamiltonian describing the Haldane model at stroboscopic
instants of time in the unitary situation is topologically non-trivial. We
establish that in the stroboscopic steady state, the system reaches a thermal
state of the Floquet Hamiltonian at a controlled temperature. Further, it is
observed that even with a coupling to a quasi-local bath, remarkably a Chern
insulator can indeed be prepared in a Chern non-trivial pure steady state which
is expected to exhibit a stroboscopic bulk-boundary correspondence. Using the
non-uniqueness of the macroscopic bulk electric polarisation of a Chern
insulator in its topological phase, we propose a generalised Chern invariant
that reflects the topology of out-of-equilibrium many-body stroboscopic states
of the Haldane model even in a dissipative ambience. The generalised topology
of dynamical Chern insulators being dependent on single-particle correlations,
is expected to manifest in experiments probing many-body quantum observables.
