Non-product property of the digital fundamental group

Banach fixed point theorem from the viewpoint of digital topology

“…To study Banach fixed point theorem for digital images, we need to recall some basic notions from digital topology such as digital k-connectivity of n-dimensional integer grids, a digital k-neighborhood, digital continuity and so forth [7,20,25,26].…”

“…"6-adjacent", "18-adjacent and "26-adjacent") well established in the context of 2-(resp. 3-)dimensional integer grids [20,25], we will say that two distinct points p, q ∈ Z n are k-(or k(t, n)-)adjacent if they satisfy the following property [7] (see also [10,11]) as follows: For a natural number t, 1 ≤ t ≤ n, two distinct points…”

“…Concretely, these k(t, n)-adjacency relations of Z n are determined according to the two numbers t, n ∈ N [7] (see also [13,14]). …”

“…Motivated by the Banach fixed point theorem [1], the other tools [5] such as a generalized Banach contraction principle [19,27] were established for studying metric spaces [18,23,24]. Digital topology has a focus on studying digital topological properties of nD digital images [25,26], which have contributed to some areas of computer sciences such as image processing, computer graphics, mathematical morphology and so forth [2,7,20]. To be specific, the notion of digital continuity was developed by Rosenfeld [26] for studying 2D and 3D digital images.…”

“…To be specific, the notion of digital continuity was developed by Rosenfeld [26] for studying 2D and 3D digital images. Then the concept was extended into the study of nD digital images, n ∈ N [7]. Up to now, several types of digital continuities have been developed [11] for studying digital images from the viewpoints of Khalimsky topology and Marcus Wyse topology and so forth [28,29].…”