Intriguing physical properties of materials stem from their chemical constituents, whereas the connection between them is often not clear. Here, we uncover a general chemical classification for the two quantum phases in the honeycomb ABX structure-topological insulator (TI) and topological Dirac semimetal (TDSM). First, we find among the 816 (existing as well as hypothetical) calculated compounds, 160 TIs (none were noted before), 96 TDSMs, 282 normal insulators (NIs), and 278 metals. Second, based on this classification, we have distilled a simple chemical regularity based on compound formulas for the selectivity between TI and TDSM: the ABX compounds that are TDSM have B atoms (part of the BX honeycomb layers) that come from the periodic table columns XI (Cu, Ag, Au) or XII (Zn, Cd, Hg), or Mg (group II), whereas the ABX compounds whose B atoms come from columns I (Li, Na, K, Rb, Cs) or II (Ca, Sr, Ba) are TIs. Third, focusing on the ABX bismide compounds that are thermodynamically stable, we find a structural motif that delivers topological insulation and stability at the same time. This study opens the way to simultaneously design new topological materials based on the compositional rules indicated here.