A digital analogue of the Jordan curve theorem (Q1827823)

The usual approach to digital topology of the plane pioneered by \textit{A. Rosenfeld} [see for example Am. Math. Mon. 86, 621--630 (1979; Zbl 0432.68061)] is to use a 4-adjacency relation for the background and an 8-adjacency relation for the image in the rectangular grid \({\mathbb Z}^ 2\). The need to use two distinct types of adjacency arises in order to be able to prove an analogue of the Jordan curve theorem for digital images. Later a purely topological theory for 2-dimensional digital spaces was proposed by \textit{E. Khalimsky, R. Kopperman} and \textit{P. R. Meyer} [Topology Appl. 36, 1--17 (1990; Zbl 0709.54017)] and extended to dimension 3 by \textit{R. Kopperman, P. R. Meyer} and the reviewer in [Discrete Comput. Geom. 6, 155--161 (1991; Zbl 0738.68086)]. The purpose of this paper is to show that similar results can be obtained using \(n\)-ary relations on \({\mathbb Z}^2\) and the (non-idempotent) closure operators associated with them. Specifically after defining a certain \(n\)-ary relation \(v_n\), it is shown that if \(J\) is a Jordan curve in \(({\mathbb Z}^2,v_n)\), then \(J\) separates \(({\mathbb Z}^2,v_n)\) into precisely two components. Whether this theory has any advantages over more traditional approaches to image processing remains to be seen.