### 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.

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.

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