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 simplicial complexes via crosscut constructions, we obtain persistence modules without requiring inclusion relations between the resulting complexes. We show that standard poset-level simplifications preserve persistent invariants and prove stability of the resulting persistence diagrams under perturbations of the input metric in a density-based instantiation.

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，我们关联一个有限拓扑的滤过结构，其结构映射为连续恒等映射。通过函子性地经由横截构造转换到偏序集和单纯复形，我们无需所得复形之间的包含关系即可获得持续模。我们证明了标准的偏序集层面简化操作能保持持续不变量，并在基于密度的实例化中，证明了所得持续图在输入度量扰动下的稳定性。