In open set recognition, deep neural networks encounter object classes that were unknown during training. Existing open set classifiers distinguish between known and unknown classes by measuring distance in a network's logit space, assuming that known classes cluster closer to the training data than unknown classes. However, this approach is applied post-hoc to networks trained with cross-entropy loss, which does not guarantee this clustering behaviour. To overcome this limitation, we introduce the Class Anchor Clustering (CAC) loss. CAC is a distance-based loss that explicitly trains known classes to form tight clusters around anchored class-dependent cluster centres in the logit space. We show that training with CAC achieves state-of-the-art open set performance for distance-based open set classifiers on the standard benchmark datasets, with a 2.4% performance increase in AUROC on the challenging TinyImageNet, without sacrificing classification accuracy. We also show that our anchored class centres achieve higher open set performance than learnt class centres, particularly on object-based datasets and large numbers of training classes.