Uncertainty is ubiquitous in geology, and efforts to characterise and communicate it are becoming increasingly important. Recent studies have quantified differences between perturbed geological models to gain insight into uncertainty. We build on this approach by quantifying differences in topology, a property that describes geological relationships in a model, introducing the concept of topological uncertainty. Data defining implicit geological models were perturbed to simulate data uncertainties, and the amount of topological variation in the resulting model suite measured to provide probabilistic assessments of specific topological hypotheses, sources of topological uncertainty and the classification of possible model realisations based on their topology. Overall, topology was found to be highly sensitive to small variations in model construction parameters in realistic models, with almost all of the several thousand realisations defining distinct topologies. In particular, uncertainty related to faults and unconformities was found to have profound topological implications. Finally, possible uses of topology as a geodiversity metric and validation filter are discussed, and methods of incorporating topological uncertainty into physical models are suggested.