We demonstrate the topological band-gap dependence of armchair honeycomb nanoribbons in a staggered sublattice potential. A scaling law is presented to quantify the band gap variation with potential strength. All armchair nanoribbons are described by one of three distinct classes depending on their width, consistent with previous classifications, namely, the well known massless Dirac condition, potentially gapless, and gapless-superlattice. The ability to tune and, in all cases close, the band-gap via external probes makes our classification particularly relevant experimentally. We propose several systems in which these results should shed considerable light, which have all already been experimentally realized.