  Non-Hermiticity can vary the topology of system, induce topological phase
transition, and even invalidate the conventional bulk-boundary correspondence.
Here, we show the introducing of non-Hermiticity without affecting the
topological properties of the original chiral symmetric Hermitian systems.
Conventional bulk-boundary correspondence holds, topological phase transition
and the (non)existence of edge states are unchanged even though the energy
bands are inseparable due to non-Hermitian phase transition. Chern number for
energy bands of the generalized non-Hermitian system in two dimension is proved
to be unchanged and favorably coincides with the simulated topological charge
pumping. Our findings provide insights into the interplay between
non-Hermiticity and topology. Topological phase transition independent of
non-Hermitian phase transition is a unique feature that beneficial for future
applications of non-Hermitian topological materials.
