In this paper, we present ThumbSpace, a software-based interaction technique that provides general one-handed thumb operation of touchscreen-based mobile devices. Our goal is to provide accurate selection of all interface objects, especially small and far targets, which are traditionally difficult to interact with using the thumb. We present the ThumbSpace design and a comparative evaluation against direct interaction for target selection. Our results show that ThumbSpace was well-received, improved accuracy for selecting targets that are out of thumb reach, and made users as effective at selecting small targets as large targets. The results further suggest user practice and design iterations hold potential to close the gap in access time between the two input methods, where ThumbSpace did not do as well as direct interaction.