Title: Homeomorphism Alignment in Two Spaces for Unsupervised Domain Adaptation. Abstract: The existing unsupervised domain adaptation methods always align the features from the source and target domains explicitly or implicitly in a common space, i.e., the domain invariant space. Explicit distribution matching always ignores the discriminability of the learned features, while implicit distribution matching such as self-supervised learning suffers from the pseudo-label noises. It is difficult to find a common space which maintains discriminative structure of the source and target domain data when aligning the data distributions. We propose a novel approach dubbed as HomeomorphisM Alignment (HMA) so that the source and target features can be aligned in two different spaces. Specifically, an invertible neural network based homeomorphism is constructed. Distribution matching method is used as a sewing up tool for connecting homeomorphism mapping between the source and target feature spaces. Theoretically, we show this mapping can preserve data topological structure, i.e., the samples in the same cluster are still in the same projected cluster. Based on this property, we adapt the model by the cross entropy of transformed and original source features and prediction consistency between target features and transformed target features. Extensive experiments demonstrate that our method can achieve the state-of-the-art results.