On storage of topological information

The usefulness of topology in science and mathematics means that topological spaces must be studied, and computers must be used in this study. Here are examples of this need from physics: In classical physics, the Euclidean spaces and compact Hausdorff spaces that arise can be approximated by finite spaces, and the goal of this paper is to discuss such approximation. A recent nonclassical development in physics uses a version of such finite approximation to view the universe as finite and eternally changing, and this is also discussed. Finite spaces are completely determined by their specialization orders. As a special case, digital n-space, used to interpret Euclidean n-space and in particular, the computer screen, is also dealt with in terms of the specialization.