Topologies associated with the one point compactifications of Khalimsky topological spaces (Q1646560)

Based on the one point compactification of the Khalimsky line (resp., the Khalimsky plane), denoted by \(({\mathbb Z}^\ast, \kappa^\ast)\) (resp., \((({\mathbb Z}^2)^\ast, (\kappa^2)^\ast)\)), the paper studies various properties of these compactifications involving the semi-\(T_{\frac{1}{2}}\) axiom, a non-Alexandroff structure, a non-cut-point space, and so on. Further, it also investigates dense subsets and nowhere dense subsets of \(({\mathbb Z}^\ast, \kappa^\ast)\) and \((({\mathbb Z}^2)^\ast, (\kappa^2)^\ast)\). The very interesting results are that the paper develops two kinds of new topologies as quotient topological spaces of \(({\mathbb Z}^\ast, \kappa^\ast)\), called an excluded two points topology and a cofinite particular point topology. These topologies can be used in pure and applied topology.