Single-basined preferences generalize single-dipped preferences by allowing for multiple worst elements. Single-dipped and single-basined preferences have played an important role in areas such as voting, strategy-proofness and matching problems. We examine the notion of single-basinedness in a choice-theoretic setting, with the set of all compact convex subsets of R-n as the domain of choice sets. In conjunction with independence of irrelevant alternatives, single-basined choice implies a structure that conforms to the motivation underlying our definition. We establish the consequences of requiring single-basined choice correspondences to be upper semicontinuous. Moreover, we extend our results to larger domains of non-convex sets.