Increasing wildfire activity globally has become an urgent issue with enormous ecological and social impacts. In this work, we focus on analyzing and quantifying the influence of landscape topology, understood as the spatial structure and interaction of multiple land-covers in an area, on fire ignition. We propose a deep learning framework, Deep Fire Topology, to estimate and predict wildfire ignition risk. We focus on understanding the impact of these topological attributes and the rationale behind the results to provide interpretable knowledge for territorial planning considering wildfire ignition uncertainty. We demonstrate the high performance and interpretability of the framework in a case study, accurately detecting risky areas by exploiting spatial patterns. This work reveals the strong potential of landscape topology in wildfire occurrence prediction and its implications to develop robust landscape management plans. We discuss potential extensions and applications of the proposed method, available as an open-source software.