Remarks on topological spaces on $ {\mathbb Z}^n $ which are related to the Khalimsky $ n $-dimensional space

<abstract><p>The present paper intensively studies various properties of certain topologies on the set of integers $ {\mathbb Z} $ (resp. $ {\mathbb Z}^n $) which are either homeomorphic or not homeomorphic to the typical Khalimsky line topology (resp. $ n $-dimensional Khalimsky topology). This finding plays a crucial role in addressing some problems which remain open in the field of digital topology.</p></abstract>

A closure operator for the digital plane

We introduce and study a closure operator on the digital plane Z2. The closure operator is shown to provide connectedness that allows for a digital analogue of the Jordan curve theorem. This enables using the closure operator for structuring the digital plane in order to study and process digital images. An advantage of the closure operator over the Khalimsky topology on Z2 is demonstrated, too.

Closure operators on graphs for modeling connectedness in digital spaces

For undirected simple graphs, we introduce closure operators on their vertex sets induced by sets of walks of the same lengths. Some basic properties of these closure operators are studied, with greater attention paid to connectedness. We focus on the closure operators induced by certain sets of walks in the 2-adjacency graph on the digital line Z, which generalize the Khalimsky topology. For the closure operators on Z2 obtained as particularly defined products of pairs of the induced closure operators on Z, we formulate and prove a digital form of the Jordan curve theorem.

Topological graphs based on a new topology on Zn and its applications

Up to now there is no homotopy for Marcus-Wyse (for short M-) topological spaces. In relation to the development of a homotopy for the category of Marcus-Wyse (for short M-) topological spaces on Z2, we need to generalize the M-topology on Z2 to higher dimensional spaces X ? Zn, n ? 3 [18]. Hence the present paper establishes a new topology on Zn; n 2 N, where N is the set of natural numbers. It is called the generalized Marcus-Wyse (for short H-) topology and is denoted by (Zn, n). Besides, we prove that (Z3, 3) induces only 6- or 18-adjacency relations. Namely, (Z3, 3) does not support a 26-adjacency, which is quite different from the Khalimsky topology for 3D digital spaces. After developing an H-adjacency induced by the connectedness of (Zn; n), the present paper establishes topological graphs based on the H-topology, which is called an HA-space, so that we can establish a category of HA-spaces. By using the H-adjacency, we propose an H-topological graph homomorphism (for short HA-map) and an HA-isomorphism. Besides, we prove that an HA-map (resp. an HA-isomorphism) is broader than an H-continuous map (resp. an Hhomeomorphism) and is an H-connectedness preserving map. Finally, after investigating some properties of an HA-isomorphism, we propose both an HA-retract and an extension problem of an HA-map for studying HA-spaces.