  We investigate gap-closings in one- and two-dimensional tight-binding models
with two bands, containing non-Hermitian hopping terms, and open boundary
conditions (OBCs) imposed along one direction. We compare the bulk OBC spectra
with the periodic boundary condition (PBC) spectra, pointing out that they do
not coincide, which is an intrinsic characteristic of non-Hermitian systems.
The non-Hermiticity, thus, results in the failure of the familiar notions of
bulk-boundary correspondence found for Hermitian systems. This necessitates the
search for topological invariants which can characterize gap-closings in open
non-Hermitian systems correctly and unambiguously. We elucidate the behaviour
of two possible candidates applicable for one-dimensional slices -- (1) the sum
of winding numbers for the two bands defined on a generalized Brillouin zone
and (2) the biorthogonal polarization (BP). While the former shows
jumps/discontinuities for some of the non-Hermitian systems studied here, at
points when an edge mode enters the bulk states and becomes delocalized, it
does not maintain quantized values in a given topological phase. On the
contrary, BP shows jumps at phase transitions, and takes the quantized value of
one or zero, which corresponds to whether an actual edge mode exists or whether
that mode is delocalized and absorbed within the bulk (not being an edge mode
anymore).
