We analyse the topology of vortex breakdown in a closed cylindrical container in the steady domain under variation of three parameters, the aspect ratio of the cylinder, the Reynolds number, and the ratio of the angular velocities of the covers. We develop a general post-processing method to obtain topological bifurcation diagrams from a database of simulations of two-dimensional flows and apply the method to axisymmetric simulations of the flow in the cylinder. Interpreting the diagrams with the aid of bifurcation theory, we obtain complete topological bifurcation diagrams for the rotation ratio in the interval [−0.04, 0.075]. In this narrow range we identify three codimension-3 bifurcation points which act as organising centres for the entire bifurcation diagram.