$\sigma$-compact Spaces in N$^{th}$-Topological Space

The principal aims of this paper include the introduction of the concept of $\sigma$-compact Spaces in $n^{th}$-topological spaces, the definition of compactness, discussion of various generalizations encompassing compactness in $n^{th}$-topological space, the properties of $\sigma$-compact Spaces. Furthermore, our study extends to the analysis of the notion of countable compactness in $n^{th}$-topological spaces. Our investigation extends to various generalizations of these spaces.