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

Abstract

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.

Topologies on Z n that Are Not Homeomorphic to the n-Dimensional Khalimsky Topological Space

“…Using a method similar to the AFPP in DTC in Remark 3, we can refer to the AFPP for simple closed curves under Kor Marcus-Wyse topology, as follows: Remark 4. In the category of K-or Marcus-Wyse topological spaces, every simple closed curve with l element does not have the AFPP (see [9,[29][30][31]).…”

“…Thus we must have f (1) = 1 and further, f (3) = 5. Owing to the mapping f (3) = 5 and SN 1 (2) = {2, 3, 5}, we obtain f (SN 1 (2)) = SN 2 ( f (2)) (see (9)). Then f −1 (SN 2 (4)) = SN 1 ( f −1 (4)) = SN 1 (2) (see (12)) so that we have f (2) = 4 and f (5) = 9.…”

“…As proven in Theorem 2, it turns out that there are countably many topologies T k , k ∈ Z \ {−1}, on Z which are not homeomorphic to the K-line topology (Z, κ). Let us now replace the subbase S k of (4) by the set S k (see (9)), where…”