Ideas from Zariski topology in the study of cubical sets, cubical maps, and their homology

Cubical sets and their homology have been used in dynamical systems as well as in digital imaging. We take a refreshing view on this topic, following Zariski ideas from algebraic geometry. The cubical topology is defined to be a topology in R in which a set is closed if and only if it is cubical. This concept is a convenient frame for describing a variety of important features of cubical sets. Separation axioms which, in general, are not satisfied here, characterize exactly those pairs of points which we want to distinguish. The noetherian property guarantees the convergence of algorithms. Moreover, maps between cubical sets which are continuous and closed with respect to the cubical topology are precisely those for whom the homology map can be defined and computed without grid subdivisions. A combinatorial version of the Vietoris-Begle is derived and used for an algorithm computing homology of maps which are continuous with respect to the Euclidean topology. ∗Research supported by a grant from NSERC †Research supported by Polish KBN grant no. 2 P03A 041 24 2000 Mathematics Subject Classification: Primary 55-04; Secondary 52B05, 54C60, 68W05, 68W30, 68U10.