ABSTRACT.

In this paper, we study functions between domain \( \zeta \)-nano topological spaces and codomain nano topological spaces, which means every nano topology has its inverse image in \( \zeta \)-nano topology (i.e., \( \zeta \)-continuous). We establish the \( \zeta \)-cluster point in \( \zeta \)-continuous. We search image from a \( \zeta \)-open set (\( \zeta \)-closed set) to a nano open set (nano closed set) is called a \( \zeta \)-open map (\( \zeta \)-closed map). Finally, some of the results are portrayed with these \( \zeta \)-continuity with \( \mathcal{N} \)-continuity and we extend to \( \zeta \)-homeomorphism.