@article{Mahdi_Hegazy_2022, title={Properties of digital spaces on \(\mathbb{Z}^{2}\)}, volume={7}, url={http://m-sciences.com/index.php/jast/article/view/194}, abstractNote={<p>The two conditions $1^d$ and $2^d$ are so that Â&nbsp;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.</p>}, number={1}, journal={Journal of Advanced Studies in Topology}, author={Mahdi, Hisham and Hegazy, Khaled}, year={2022}, month={Jun.}, pages={45–53} }