The covering-based rough set (CBRS) theory provides a flexible framework for data approximation and classification under uncertainty, where its effectiveness critically depends on the construction of coverings. Traditional approaches employing the Mapper algorithm often suffer from instability and parameter sensitivity due to their reliance on projections and clustering. This study establishes a novel direction for geometric rough set modeling with topological constraints by proposing a topology-driven classification framework that integrates the Ball Mapper algorithm into the CBRS theory, known as the Ball Mapper-based rough set (BMbRS). Furthermore, it enables the construction of projection-free coverings based solely on metric geometry. Within the BMbRS model, we examine all derived coverings in order to obtain a more comprehensive view of the underlying data structure, allowing us to identify the most suitable covering configuration for a given classification task. An experimental evaluation across five benchmark healthcare datasets demonstrates the advantages of the proposed approach. It achieves high average classification quality, with scores of 0.910 (Diabetes), 0.967 (Heart disease), 0.943 (Lung cancer), 0.971 (Fetal health), and 0.962 (Breast cancer). These outcomes indicate that the novel approach offers better interpretability and stability while maintaining a manageable number of parameters, especially in comparison with conventional Mapper-based methods.