  By using the self-consistent Born approximation, we investigate disorder
effect induced by short-range impurities on the band-gap of a seminal
two-dimensional (2D) system, whose phase diagram contains trivial,
single-band-inverted and double-band-inverted states. Following the
density-of-states (DOS) evolution, we demonstrate multiple closings and
openings of the band-gap with the increase of the disorder strength.
Calculations of the spectral function describing the quasiparticles at the
$\Gamma$ point of the Brillouin zone evidence that the observed band-gap
behavior is unambiguously caused by the topological phase transitions due to
the mutual inversions between the first and second electron-like and hole-like
subbands. We also find that an increase in the disorder strength in the
double-inverted state always leads to the band-gap closing due to the overlap
of the tails of DOS from conduction and valence subbands.
