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

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 Z2 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 l1=l2, where SCMAl means a simple closed Marcus Wyse adjacent (for brevity, MA-) curve with l elements in Z2. 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,γX) with MA-adjacency and morphisms are all M-maps f:(X,γX)→(Y,γY) for every ordered pair of objects (X,γX) and (Y,γY), and DTC(4) is the category whose objects are digital images (X,k) in Z2 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.