Sleep-stage and apnea-hypopnea index (AHI) are the most important metrics in the diagnosis of sleep syndrome disease. In previous studies, these two tasks are usually implemented separately, which is both time- and resource-consuming. In this work, we propose a novel single electroencephalogram (EEG)-based collaborative learning network (EEG-CLNet) for simultaneous sleep staging and obstructive sleep apnea (OSA) event detection through multitask collaborative learning. The EEG-CLNet regards different tasks as a common unit to extract features from intragroups via both local parameter sharing and cross-task knowledge distillation (CTKD), rather than just sharing parameters or shortening the distance between different tasks. Our approach has been validated on two datasets with the same or better performance than other methods. The experimental results show that our method achieves a performance gain of 1%-5% compared with the baseline. Compared to previous works where two or even more models were required to perform sleep staging and OSA event detection, the EEG-CLNet could reduce the total number of model parameters and facilitate the model to mine the hidden relationships between different task semantic information. More importantly, it effectively alleviates the task bias problem in hard parameter sharing. As a consequence, this approach has notable potential to be a solution for a lightweight wearable sleep monitoring system in the future.