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Local discriminative representation is needed in many medical image analysis tasks such as identifying sub-types of lesion or segmenting detailed components of anatomical structures. However, the commonly applied supervised representation learning methods require a large amount of annotated data, and unsupervised discriminative representation learning distinguishes different images by learning a global feature, both of which are not suitable for localized medical image analysis tasks. In order to avoid the limitations of these two methods, we introduce local discrimination into unsupervised representation learning in this work. The model contains two branches: one is an embedding branch which learns an embedding function to disperse dissimilar pixels over a low-dimensional hypersphere; and the other is a clustering branch which learns a clustering function to classify similar pixels into the same cluster. These two branches are trained simultaneously in a mutually beneficial pattern, and the learnt local discriminative representations are able to well measure the similarity of local image regions. These representations can be transferred to enhance various downstream tasks. Meanwhile, they can also be applied to cluster anatomical structures from unlabeled medical images under the guidance of topological priors from simulation or other structures with similar topological characteristics. The effectiveness and usefulness of the proposed method are demonstrated by enhancing various downstream tasks and clustering anatomical structures in retinal images and chest X-ray images.