The physical origin of the fractal topological band-gap structure in the one-dimensional Thue–Morse system has been revealed, which is characterized by the evolutions of two types of topological singularities with zero-scattering properties and the paths of phase vortex points, which are the mirrored paths of the first-type singularities. The field distribution of the upper and lower gap-edge states will interchange when the traditional gaps are closed and reopened. The topologically protected edge-states are found at both traditional gaps and fractal gaps. Our work broadens the topological properties of quasicrystals or aperiodic systems and provides potential applications in new optoelectronic devices.