Topologies associated with the one point compactifications of Khalimsky topological spaces

Abstract

In this paper, after discussing the one point compactification of the Khalimsky line (resp. the Khalimsky plane), denoted by (Z⁎,κ⁎) (resp. ((Z2)⁎,(κ2)⁎)), we study various properties of these compactifications associated with the semi-T12 axiom, a non-Alexandroff structure, a non-cut-point space and so forth. We also investigate dense subsets and nowhere dense subsets of (Z⁎,κ⁎) and ((Z2)⁎,(κ2)⁎). Finally, motivated by a particular point topology and an excluded point topology, we develop two kinds of new topologies as quotient topological spaces of (Z⁎,κ⁎) called an excluded two points topology and a cofinite particular point topology.