Structure theorem
Structure theorem [ edit ] The structure theorem is of central importance to TDA; as commented by G. Carlsson, "what makes homology useful as a discriminator between topological spaces is the fact that there is a classification theorem for finitely generated abelian groups." [3] (see the fundamental theorem of finitely generated abelian groups ). The main argument used in the proof of the original structure theorem is the standard structure theorem for finitely generated modules over a principal ideal domain . [9] However, this argument fails if the indexing set is ( � , ≤ ) . [3] In general, not every persistence module can be decomposed into intervals. [70] Many attempts have been made at relaxing the restrictions of the original structure theorem. [ clarification needed ] The case for pointwise finite-dimensional persistence modules indexed by a locally finite subset of � is solved based on the work of Webb. [71] The most notable result is done by Crawley-Boeve