Homotopy, homology, and persistent homology using closure spaces and filtered closure spaces

We develop persistent homology in the setting of filtered (Cech) closure spaces. Examples of filtered closure spaces include filtered topological spaces, metric spaces, weighted graphs, and weighted directed graphs. We use various products and intervals for closure spaces to obtain six homotopy theories, six cubical singular homology theories and three simplicial singular homology theories. Applied to filtered closure spaces, these homology theories produce persistence modules. We extend the definition of Gromov-Hausdorff distance to filtered closure spaces and use it to prove that these persistence modules and their persistence diagrams are stable. We also extend the definitions of Vietoris-Rips and Cech complexes to give functors on closure spaces and prove that their persistent homology is stable. The Vietoris-Rips functor has a left adjoint, which is defined using stars of simplices; in contrast the Cech functor does not have a left or right adjoint.Comment: 56 pages, extensive improvements to the exposition, some results adde