Digital Jordan curves (Q2433687)

An Alexandroff topology is defined on the set \(\mathbb Z^{2}\) of integer lattice points of the Euclidean plane \(\mathbb R^{2}\). This topology has the advantage over the usual Khalimsky topology on \(\mathbb Z^{2}\) in that any cycle of a certain square-diagonal graph of type 4 is a digital Jordan curve in the comparability graph induced by the specialization order (defined by \(x\leq y\) if and only if \(x\in\text{ cl}(y)\)) of the topology. However, the topology is not the product of two topologies on \(\mathbb Z\).