Recently, clustering-based methods have been the dominant solution for unsupervised person re-identification (ReID). Memory-based contrastive learning is widely used for its effectiveness in unsupervised representation learning. However, we find that the inaccurate cluster proxies and the momentum updating strategy do harm to the contrastive learning system. In this paper, we propose a real-time memory updating strategy (RTMem) to update the cluster centroid with a randomly sampled instance feature in the current mini-batch without momentum. Compared to the method that calculates the mean feature vectors as the cluster centroid and updating it with momentum, RTMem enables the features to be up-to-date for each cluster. Based on RTMem, we propose two contrastive losses, i.e., sample-to-instance and sample-to-cluster, to align the relationships between samples to each cluster and to all outliers not belonging to any other clusters. On the one hand, sample-to-instance loss explores the sample relationships of the whole dataset to enhance the capability of density-based clustering algorithm, which relies on similarity measurement for the instance-level images. On the other hand, with pseudo-labels generated by the density-based clustering algorithm, sample-to-cluster loss enforces the sample to be close to its cluster proxy while being far from other proxies. With the simple RTMem contrastive learning strategy, the performance of the corresponding baseline is improved by 9.3% on Market-1501 dataset. Our method consistently outperforms state-of-the-art unsupervised learning person ReID methods on three benchmark datasets. Code is made available at:https://github.com/PRIS-CV/RTMem.