Rashba spin orbit torque derived from the broken inversion symmetry at ferromagnet/heavy metal interfaces has potential application in spintronic devices. In conventional description of the precessional and damping components of the Rashba spin orbit torque in magnetization textures, the decomposition coefficients are assumed to be independent of the topology of the underlying structure. Contrary to this common wisdom, for Schrödinger electrons trespassing ballistically across a magnetic domain wall, we found that the decomposition coefficient of the damping component is determined by the topology of the domain wall. The resultant damping Rashba spin orbit torque is protected by the topology of the underlying magnetic domain wall and robust against small deviations from the ideal domain wall profile. Our identification of a topological damping Rashba spin orbit torque component in magnetic domain walls will help to understand experiments on current driven domain wall motion in ferromagnet/heavy metal systems with broken inversion symmetry and to facilitate its utilization in innovative device designs.One main theme in the field of nanomagnetism is to search for new approaches to realize fast and energy efficient manipulation of magnetic state, rather than using the conventional magnetic field. In the past three decades, several promising candidates, such as electric field 1 , laser pulses 2 and spin current through the spin transfer torque (STT) 3-6 , were proposed. A recent development along this line is the emergence of the Rashba spin orbit torque (RSOT) in magnetic systems without inversion symmetry. In a simple picture 7 , the electric field along the symmetry breaking direction is equivalent to a magnetic field, dubbed the Rashba field, in the rest reference frame of an electron in motion. Due to the s-d exchange between the local and itinerant spin degrees of freedom, the Rashba field is transformed into the RSOT acting on the local magnetization.When it was first proposed, only the precessional component 8-11 of the RSOT, corre-