  The interplay between dissipation and correlation can lead to new emergent
phenomena. Here we study non-equilibrium phases of matter with robust
topological degeneracy of steady states, which is a generalization of the
ground-state topological degeneracy of closed systems. Specifically, we
construct two representative Lindbladians using engineered dissipation, and
exactly solve the steady states with topological degeneracy. We find that while
the degeneracy is fragile under noise in two dimensions, it is stable in three
dimensions, where a genuine many-body phase with topological degeneracy is
realized. We identify universal features of dissipative topological physics
such as the deconfined emergent gauge field and slow relaxation dynamics of
topological defects. The transition from a topologically ordered phase to a
trivial phase is also investigated via numerical simulation. Our work
highlights the essential difference between ground-state topological order in
closed systems and steady-state topological order in open systems.
