Differentiable Structures of the Induced Topology on a Manifold
A manifold, to be differentiable, we need the structures to be 𝐂 ∞. It ensures that the transition maps are smooth by showing the coordinate representatives of the manifold are homeomorphisms and their images are open in 𝐑 𝐧 . In this study, we see that the coordinate representatives of the induced topology on a manifold are also homeomorphisms and their images are also open in 𝐑 𝐧 . Therefore, they have the same 𝐂 ∞-structure as the manifold. In this paper, some aspects of sets, functions, topology and differentiable properties of manifolds have been studied. A relation between the differentiable structures of the induced topology on a manifold relative to the topology of the manifold have been established by using various definitions, theorems, corollaries and examples.