Finiteness of rank invariants of multidimensional persistent homology groups

Persistent topology is a theory for studying objects related to computer vision and computer graphics. It involves analyzing the qualitative and quantitative behavior of real-valued functions defined over topological spaces. This is achieved by considering the filtration obtained from the sequence of nested lower level sets of the function under study, and by encoding the scale at which a topological feature (e.g., a connected component, a tunnel, a void) is created, and when it is annihilated along this filtration. In this framework, multidimensional persistent homology groups capture the homology of a multi-parameter increasing family of spaces. For application purposes, these groups are further encoded by simply considering their rank, which yields a parameterized version of Betti numbers, called rank invariants . In this note we give a sufficient condition for their finiteness. This condition is sharp for spaces embeddable in the euclidean n-dimensional space .