We propose a functorial framework for persistent homology based on finite topological spaces and their associated posets. Starting from a finite metric space, we associate a filtration of finite topologies whose structure maps are continuous identity maps. By passing functorially to posets and to order complexes, we obtain persistence modules without requiring inclusion relations between the resulting complexes. We show that standard poset-level simplifications preserve persistent invariants and establish stability of the resulting persistence diagrams under perturbations of the input metric in a basic density-based instantiation, illustrating how stability arguments arise naturally in our framework. We further introduce a concrete density-guided construction, designed to be faithful to anchor neighborhood structure at each scale, and demonstrate its practical viability through an implementation tested on real datasets.

翻译：我们提出了一种基于有限拓扑空间及其关联偏序集的函子化持续同调框架。从有限度量空间出发，我们构建有限拓扑的滤过结构，其结构映射为连续恒等映射。通过函子化地转换到偏序集及其序复形，我们无需所得复形间的包含关系即可获得持续模。我们证明了标准偏序集层面的简化操作能保持持续不变量，并在基于密度的基础实例化中，建立了所得持续图在输入度量扰动下的稳定性，从而阐明稳定性论证如何在本框架中自然产生。我们进一步提出了一种具体的密度引导构造方法，该方法旨在忠实反映各尺度下的锚点邻域结构，并通过在真实数据集上测试的实现结果验证了其实际可行性。