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Unsupervised graph representation learning has emerged as a powerful tool to address real-world problems and achieves huge success in the graph learning domain. Graph contrastive learning is one of the unsupervised graph representation learning methods, which recently attracts attention from researchers and has achieved state-of-the-art performances on various tasks. The key to the success of graph contrastive learning is to construct proper contrasting pairs to acquire the underlying structural semantics of the graph. However, this key part is not fully explored currently, most of the ways generating contrasting pairs focus on augmenting or perturbating graph structures to obtain different views of the input graph. But such strategies could degrade the performances via adding noise into the graph, which may narrow down the field of the applications of graph contrastive learning. In this paper, we propose a novel graph contrastive learning method, namely \textbf{D}ual \textbf{S}pace \textbf{G}raph \textbf{C}ontrastive (DSGC) Learning, to conduct graph contrastive learning among views generated in different spaces including the hyperbolic space and the Euclidean space. Since both spaces have their own advantages to represent graph data in the embedding spaces, we hope to utilize graph contrastive learning to bridge the spaces and leverage advantages from both sides. The comparison experiment results show that DSGC achieves competitive or better performances among all the datasets. In addition, we conduct extensive experiments to analyze the impact of different graph encoders on DSGC, giving insights about how to better leverage the advantages of contrastive learning between different spaces.