Good Pairs of Adjacency Relations in Arbitrary Dimensions

TL;DR

This paper explores the concept of 'good pairs' of adjacency relations in digital topology across arbitrary dimensions, providing methods to select appropriate cubical adjacencies and extending known results to higher dimensions.

Contribution

It demonstrates how to choose cubical adjacencies in arbitrary dimensions to form good pairs and offers a new proof that Khalimsky-topology implies good pairs, extending existing theory.

Findings

Identification of models for good pairs in arbitrary dimensions

Methods to select cubical adjacencies for good pairs

Extension of known results to higher dimensions

Abstract

In this text we show, that the notion of a "good pair" that was introduced in the paper "Digital Manifolds and the Theorem of Jordan-Brouwer" has actually known models. We will show, how to choose cubical adjacencies, the generalizations of the well known 4- and 8-neighborhood to arbitrary dimensions, in order to find good pairs. Furthermore, we give another proof for the well known fact that the Khalimsky-topology implies good pairs. The outcome is consistent with the known theory as presented by T.Y. Kong, A. Rosenfeld, G.T. Herman and M. Khachan et.al and gives new insights in higher dimensions.