We explore the concept of Euclidean locality within treonic topological spaces. Our study establishes a foundational theoretical framework, elucidating the properties of continuity, homeomorphisms, and compactness in these spaces. We assess the Euclidean locality of treonic spaces through the analysis of specific homeomorphisms, which enables us to define manifolds in treonic spaces and extend recent research on Bermejo Algebras. For the first time, we characterize treonic manifolds by incorporating the property of Euclidean locality alongside previously studied properties such as Hausdorff spaces and second countable spaces. Our findings advance the understanding of the topological and geometric properties of treonic spaces, providing significant insights for advanced mathematical research.