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Nuclei segmentation is a crucial task for whole slide image analysis in digital pathology. Generally, the segmentation performance of fully-supervised learning heavily depends on the amount and quality of the annotated data. However, it is time-consuming and expensive for professional pathologists to provide accurate pixel-level ground truth, while it is much easier to get coarse labels such as point annotations. In this paper, we propose a weakly-supervised learning method for nuclei segmentation that only requires point annotations for training. First, coarse pixel-level labels are derived from the point annotations based on the Voronoi diagram and the k-means clustering method to avoid overfitting. Second, a co-training strategy with an exponential moving average method is designed to refine the incomplete supervision of the coarse labels. Third, a self-supervised visual representation learning method is tailored for nuclei segmentation of pathology images that transforms the hematoxylin component images into the H&E stained images to gain better understanding of the relationship between the nuclei and cytoplasm. We comprehensively evaluate the proposed method using two public datasets. Both visual and quantitative results demonstrate the superiority of our method to the state-of-the-art methods, and its competitive performance compared to the fully-supervised methods. Code: https://github.com/hust-linyi/SC-Net