In open-set recognition (OSR), a promising strategy is exploiting pseudo-unknown data outside given K known classes as an additional K+1-th class to explicitly model potential open space. However, treating unknown classes without distinction is unequal for them relative to known classes due to the category-agnostic and scale-agnostic of the unknowns. This inevitably not only disrupts the inherent distributions of unknown classes but also incurs both class-wise and instance-wise imbalances between known and unknown classes. Ideally, the OSR problem should model the whole class space as K+∞, but enumerating all unknowns is impractical. Since the core of OSR is to effectively model the boundaries of known classes, this means just focusing on the unknowns nearing the boundaries of targeted known classes seems sufficient. Thus, as a compromise, we convert the open classes from infinite to K, with a novel concept Target-Aware Universum (TAU) and propose a simple yet effective framework Dual Contrastive Learning with Target-Aware Universum (DCTAU). In details, guided by the targeted known classes, TAU automatically expands the unknown classes from the previous 1 to K, effectively alleviating the distribution disruption and the imbalance issues mentioned above. Then, a novel Dual Contrastive (DC) loss is designed, where all instances irrespective of known or TAU are considered as positives to contrast with their respective negatives. Experimental results indicate DCTAU sets a new state-of-the-art.