  We introduce a classification scheme for symmetry protected topological
phases applicable to stationary states of open systems based on a
generalization of the many-body polarization. The polarization can be used to
probe the topological properties of non-interacting and interacting closed and
open systems as well and remains a meaningful quantity even in the presence of
moderate particle-number fluctuations. As examples, we discuss two open-system
versions of a topological Thouless pump in the steady state of one-dimensional
lattices driven by Markovian reservoirs. In the analogous unitary system, the
Rice-Mele model, symmetries enforce topological properties which lead to a
non-trivial winding of the geometric Zak phase upon cyclic variations of model
parameters. Associated with this is a winding of the many-body polarization,
corresponding to a quantized transport in the bulk (Thouless pump). We here
show that in the open system, where the Zak phase looses its meaning, the same
symmetries enforce a winding of the generalized many- body polarization. This
winding is shown to be robust against Hamiltonian perturbations as well as
homogeneous dephasing and particle losses..
