The state of polarization of strongly focused, radially polarized electromagnetic fields is examined. It is found that several types of polarization singularities exist. Their relationship is investigated, and it is demonstrated that on smoothly varying a system parameter, such as the aperture angle of the lens, different polarization singularities can annihilate each other. For example, the evolution of a lemon into a monstar and its subsequent annihilation with a star is studied. Also, the quite rare collision of a C-line and an L -line, resulting in a V-point, is observed.