Properties of digital spaces on \(\mathbb{Z}^{2}\)

Abstract

The two conditions $1^d$ and $2^d$ are so that any digital topology on \(\mathbb{Z}^d\) satisfies them is topologically connected whenever it is graphically connected. In this paper, we prove that the digital topologies on \(\mathbb{Z}^d\) are \(g\)-locally finite \(T_0\) Alexandroff spaces. We study the properties of the two digital topologies on \(\mathbb{Z}^2\) that satisfy \(1^2\) and \(2^2\). We describe the specialization orders of these topologies, and we determine the points in \(\mathbb{Z}^2\) that are minimal, maximal, and saddle points. We prove that, the summation topology is homeomorphic to the Khalimsky topology.