While conventional computer vision emphasizes pixel-level and feature-based objectives, medical image analysis of intricate biological structures necessitates explicit representation of their complex topological properties. Despite their successes, deep learning models often struggle to accurately capture the connectivity and continuity of fine, sometimes pixel-thin, yet critical structures due to their reliance on implicit learning from data. To address this challenge, we introduce Conformable Convolution, a novel convolutional layer designed to explicitly impose topological consistency. Conformable Convolution learns adaptive kernel offsets that focus on regions of high topological significance within an image. This prioritization is guided by our proposed Topological Posterior Generator (TPG) module, which leverages persistent homology. The TPG module identifies key topological features and guides the convolutional layers by applying persistent homology to feature maps transformed into cubical complexes. Unlike existing approaches that are merely aware of topology, our method explicitly constrains the learning process to ensure topological correctness. The proposed modules are architecture-agnostic, enabling them to be integrated seamlessly into various architectures. We showcase the effectiveness of our framework in the segmentation task, where preserving the interconnectedness of structures is critical. The results on three diverse datasets demonstrate that our framework effectively preserves the topology both quantitatively and qualitatively. The source code of this work will be published upon its acceptance.