We argue that semiconducting quenched liquids and frozen glasses may exhibit a set of peculiar electronic states of topological origin. These states reside at strained regions arising during structural reconfigurations between distinct aperiodic states intrinsic to quenched melts. The strained regions are domain walls separating the distinct aperiodic states; their number is about 10(20) cm(-3) in all glassformers owing to the universal dynamics of deeply supercooled melts. Even though located near the middle of the forbidden gap, the topological states are rather extended in one direction while being centered at under- and overcoordinated atoms. The states exhibit the reverse charge-spin relation, the majority of states being diamagnetic and charged. The topological states may be sufficient to account for a number of irradiation-induced phenomena in amorphous semiconductors, including electron spin resonance signal, midgap absorption, photoluminescence, and the fatigue of photoluminescence. We propose experiments to test the present microscopic picture.