Definition (Compatible)

Two charts $(U,\phi : U → \mathbb{R}^{n}), (V, ψ : V → \mathbb{R}^{n})$ of a Topological Manifold are $C^{\infty}$-compatible if the two maps $$\begin{align*} φ ◦ ψ^{-1} : ψ(U ∩V ) → \phi (U ∩V )\\ ψ ◦ \phi^{-1} : \phi (U ∩V ) → ψ(U ∩V ) \end{align*}$$ are Smooth. We call these maps transition functions.