Excellent performance has been achieved on semi-supervised medical image segmentation, but existing algorithms perform relatively poorly for objects with variable topologies and weak boundaries. In this paper, we propose a novel cross co-teaching framework, called Cross-structure-task Collaborative Teaching (CroCT), which not only can effectively handle variable topologies, but also strengthens the learning for weak boundaries of unlabeled data. Specifically, a new cross-structure-task collaborative teaching mechanism is developed based on our designed “E-Net” structure composed of a shared encoder and two decoder branches with distinct learning paradigms, which asks these two branches to regress topology-aware signed distance functions and densely-predicted segmentation masks for each other. Powered by the collaboration across different structural biases and sequence-related tasks, our CroCT can extract more discriminative yet complementary representations from abundant raw medical data to promote the consistency learning generalization, further boosting the performance for tackling highly diverse shapes and topological changes intra-/inter-slices. Besides, it guarantees the diversities from multi-levels, i.e., structure and task perspectives, to preclude prediction uncertainty. In addition, a novel adaptive boundary enhancing (ABE) module is proposed to introduce compact annularly enhanced boundary features into semi-supervised training, which significantly improves weak boundary perception ability for unlabeled data while facilitating collaborative teaching for efficiently propagating complementary knowledge across different branches. The extensive experiments on three challenging medical benchmarks, employing different labeled settings, demonstrate the superiority of our CroCT over recent state-of-the-art competitors.