The steady-state flow generated by a rotating bottom in a closed cylindrical container and the resulting vortex breakdown bubbles have been studied experimentally. By comparing the flow inside two different container geometries, one with a rigid cover and the other with a free surface, we examined the way in which the formation and structure of the breakdown bubbles depend on the surrounding flow. Details of the flow were visualized by means of the electrolytic precipitation technique, whereas a particle tracking technique was used to characterize the whole flow field. We found that the breakdown bubbles inside the container flow are in many ways similar to those in vortex tubes. First, the bubbles are open with in- and outflow and second, their structure is, like in the case of vortex breakdown in pipe flows, highly axisymmetric on the upstream side of the bubble and asymmetric on their rear side. However, and surprisingly, we observed bubbles which are open and stationary at the same time. This shows that open breakdown bubbles are not necessarily the result of periodic oscillations of the recirculation zone. The asymmetry of the flow structure is found to be related to the existence of asymmetric flow separations on the container wall. If the angular velocity of the rotating bottom is increased the evolution of the breakdown bubbles is different in both configurations: in the rigid cover case the breakdown bubbles disappear but persist in the free surface case.