A 3D digital Jordan–Brouwer separation theorem

Abstract

We introduce and discuss a concept of connectedness induced by an n-ary relation ( $$n>1$$ an integer). In particular, for every integer $$n>1$$ , we define an n-ary relation $$R_n$$ on the digital line $$\mathbb {Z}$$ and equip the digital space $$\mathbb {Z}^3$$ with the n-ary relation $$R_n^3$$ obtained as a special product of three copies of $$R_n$$ . For $$n=2$$ , the connectedness induced by $$R_n^3$$ coincides with the connectedness given by the Khalimsky topology on $$\mathbb {Z}^3$$ and we show that, for every integer $$n>2$$ , it allows for a digital analog of the Jordan–Brouwer separation theorem for three-dimensional spaces. An advantage of the connectedness induced by $$R_n^3$$ ( $$n>2$$ ) over that given by the Khalimsky topology is shown, too.