The semi-$T_3$-separation axiom of Khalimsky topological spaces

Abstract

The paper initially studies both the $s$-$T_3$-separation and the semi-$T_3$-separation axiom of Khalimsky ($K$- for brevity) topological spaces.
   To do this work, we first investigate some properties of semi-open and semi-closed sets with respect to the operations of the union or intersection and further, a homeomorphism, and a semi-homeomorphism.
   Next, we study various properties of semi-topological properties of $K$-topological spaces such as simple $K$-paths.
   Finally, after introducing the notion of a semi-$T_3$-separation axiom which is broader than the $s$-$T_3$-separation axiom,
    we find the sufficient and necessary condition for a Khalimsky topological space to satisfy the semi-$T_3$-separation axiom.