TY - JOUR
AU - Mahdi, Hisham
AU - Hegazy, Khaled
PY - 2022/06/01
Y2 - 2022/08/19
TI - Properties of digital spaces on \(\mathbb{Z}^{2}\)
JF - Journal of Advanced Studies in Topology
JA - J. of Adv. Stud. in Top.
VL - 7
IS - 1
SE - Research Articles
DO -
UR - http://m-sciences.com/index.php/jast/article/view/194
SP - 45–53
AB - <p>The two conditions $1^d$ and $2^d$ are so that Â 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>
ER -