Polymer networks can have a range of desirable properties such as mechanical strength, wide compositional diversity between different materials, permanent porosity, convenient processability and broad solvent compatibility(1,2). Designing polymer networks from the bottom up with new structural motifs and chemical compositions can be used to impart dynamic features such as malleability or self-healing, or to allow the material to respond to environmental stimuli(3-8). However, many existing systems exhibit only one operational state that is defined by the material's composition and topology(3-6); or their responsiveness may be irreversible(7,9,10) and limited to a single network property(11,12) (such as stiffness). Here we use cooperative self-assembly as a design principle to prepare a material that can be switched between two topological states. By using networks of polymer-linked metal-organic cages in which the cages change shape and size on irradiation, we can reversibly switch the network topology with ultraviolet or green light. This photoswitching produces coherent changes in several network properties at once, including branch functionality, junction fluctuations, defect tolerance, shear modulus, stress-relaxation behaviour and self-healing. Topology-switching materials could prove useful in fields such as soft robotics and photo-actuators as well as providing model systems for fundamental polymer physics studies.