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Out-of-distribution (OOD) detection is the key to deploying models safely in the open world. For OOD detection, collecting sufficient in-distribution (ID) labeled data is usually more time-consuming and costly than unlabeled data. When ID labeled data is limited, the previous OOD detection methods are no longer superior due to their high dependence on the amount of ID labeled data. Based on limited ID labeled data and sufficient unlabeled data, we define a new setting called Weakly-Supervised Out-of-Distribution Detection (WSOOD). To solve the new problem, we propose an effective method called Topological Structure Learning (TSL). Firstly, TSL uses a contrastive learning method to build the initial topological structure space for ID and OOD data. Secondly, TSL mines effective topological connections in the initial topological space. Finally, based on limited ID labeled data and mined topological connections, TSL reconstructs the topological structure in a new topological space to increase the separability of ID and OOD instances. Extensive studies on several representative datasets show that TSL remarkably outperforms the state-of-the-art, verifying the validity and robustness of our method in the new setting of WSOOD.