  We develop persistent homology in the setting of filtrations of (Cech)
closure spaces. Examples of filtrations of closure spaces include metric
spaces, weighted graphs, weighted directed graphs, and filtrations of
topological spaces. 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 filtrations of closure
spaces, these homology theories produce persistence modules. We extend the
definition of Gromov-Hausdorff distance from metric spaces to filtrations of
closure spaces and use it to prove that any persistence module obtained from a
homotopy-invariant functor on closure spaces is stable.
