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Open-set domain adaptation (OSDA) has gained considerable attention in many visual recognition tasks. However, most existing OSDA approaches are limited due to three main reasons, including: (1) the lack of essential theoretical analysis of generalization bound, (2) the reliance on the coexistence of source and target data during adaptation, and (3) failing to accurately estimate the uncertainty of model predictions. We propose a Progressive Graph Learning (PGL) framework that decomposes the target hypothesis space into the shared and unknown subspaces, and then progressively pseudo-labels the most confident known samples from the target domain for hypothesis adaptation. Moreover, we tackle a more realistic source-free open-set domain adaptation (SF-OSDA) setting that makes no assumption about the coexistence of source and target domains, and introduce a balanced pseudo-labeling (BP-L) strategy in a two-stage framework, namely SF-PGL. Different from PGL that applies a class-agnostic constant threshold for all target samples for pseudo-labeling, the SF-PGL model uniformly selects the most confident target instances from each category at a fixed ratio. The confidence thresholds in each class are regarded as the 'uncertainty' of learning the semantic information, which are then used to weigh the classification loss in the adaptation step. We conducted unsupervised and semi-supervised OSDA and SF-OSDA experiments on the benchmark image classification and action recognition datasets. Additionally, we find that balanced pseudo-labeling plays a significant role in improving calibration, which makes the trained model less prone to over-confident or under-confident predictions on the target data. Source code is available at https://github.com/Luoyadan/SF-PGL.