Consider for simplicity a two-band system in one spatial dimension, if by tuning some parameters of the system, the band structure changes as follows(vertical axis as energy, horizontal axis as lattice momentum)
Then does it count as a "band gap closing"? Normally I would say yes, because the spectrum (range of permissible energies) changes from two disjoint intervals to one interval.
Now what confuses me is that, in topological-insulator context I've seen, people often say "Chern numbers cannot change without closing some band gap", but in the above illustrated band-closing situation, Chern numbers seem to be well-defined all the time since the two band curves don't intersect, and so they must stay constant. So is it a slight abuse of language of "Chern number cannot change without closing the band gap" where they actually mean “Chern numbers cannot change without some band curves intersecting”, or is there something I misunderstood so that the example I showed does not count as bang gap closing?