εε-connectedness, finite approximations, shape theory and coarse graining in hyperspaces

We use upper semifinite hyperspaces of compacta to describe εε-connectedness and to compute homology from finite approximations. We find a new connection between εε-connectedness and the so-called Shape Theory  . We construct a geodesically complete RR-tree, by means of εε-components at different resolutions, whose behavior at infinite captures the topological structure of the space of components of a given compact metric space. We also construct inverse sequences of finite spaces using internal finite approximations of compact metric spaces. These sequences can be converted into inverse sequences of polyhedra and simplicial maps by means of what we call the Alexandroff–McCord correspondence. This correspondence allows us to relate upper semifinite hyperspaces of finite approximation with the Vietoris–Rips complexes of such approximations at different resolutions. Two motivating examples are included in the introduction. We propose this procedure as a different mathematical foundation for problems on data analysis. This process is intrinsically related to the methodology of shape theory. This paper reinforces Robins’s idea of using methods from shape theory to compute homology from finite approximations.