Topological digital topology

Abstract

The usefulness of topology in science and mathematics means that topological spaces must be studied, and computers should be used in this study. We discuss how many useful spaces (including all compact Hausdorff spaces) can be approximated by finite spaces, and these 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. Indeed, algorithms written using the specialization are comparable in difficulty, storage usage and speed to those which use the traditional (8,4), (4,8) and (6,6) adjacencies, and are of course completely representative of the spaces. © Springer-Verlag Berlin Heidelberg 2003.