In recent years gaze estimation methods have made substantial progress, driven by the numerous application areas including human-robot interaction, visual attention estimation and foveated rendering for virtual reality headsets. However, many gaze estimation methods typically assume that the subject's eyes are open; for closed eyes, these methods provide irregular gaze estimates. Here, we address this assumption by first introducing a new open-sourced dataset with annotations of the eye-openness of more than 200,000 eye images, including more than 10,000 images where the eyes are closed. We further present baseline methods that allow for blink detection using convolutional neural networks. In extensive experiments, we show that the proposed baselines perform favourably in terms of precision and recall. We further incorporate our proposed RT-BENE baselines in the recently presented RT-GENE gaze estimation framework where it provides a real-time inference of the openness of the eyes. We argue that our work will benefit both gaze estimation and blink estimation methods, and we take steps towards unifying these methods.