Generalizations of continuity of maps and homeomorphisms for studying 2D digital topological spaces and their applications

摘要

The present paper establishes two new maps such as an M-map and an M-isomorphism which are generalizations of a Marcus Wyse (for brevity, M-) continuous map and an M-homeomorphism because an M-continuous map is so rigid that some geometric transformations are not M-continuous maps (see Remark 3.2). Furthermore, it proves that in Z(2) an M-map and an M-isomorphism are equivalent to a (digitally) 4-continuous map and a (digitally) 4-isomorphism, respectively. Besides, the paper proves that SCMAl1 is M-isomorphic to SCMAl2 if and only if l(1) = l(2), where SCMAl means a simple closed Marcus Wyse adjacent (for brevity, MA-) curve with 1 elements in Z(2). Finally, the paper proves that MAC is equivalent to DTC(4) (see Theorem 6.7), where MAC is the category whose objects are M-topological spaces (X, gamma(X)) with MA-adjacency and morphisms are all M-maps f: (X,gamma(X)) -> (Y, gamma(Y)) for every ordered pair of objects (X,gamma(X)) and (Y, gamma(Y)), and DTC(4) is the category whose objects are digital images (X, k) in Z(2) and morphisms are (digitally) 4-continuous maps. Besides, we propose the notion of an MA-retract for compressing 2D digital spaces. Using this new approach, we can substantially study and classify 2D digital topological spaces and 2D digital images. (C) 2015 Elsevier B.V. All rights reserved.