Definition:Differentiable Structure

Definition
Let $M$ be a locally Euclidean space of dimension $d$.

Then a differentiable structure of class $C^k$ $\mathscr F$ on $M$ is a collection of co-ordinate systems $\{(U_\alpha,\phi_\alpha) : \alpha \in A\}$ such that:


 * $(1): \quad \displaystyle \bigcup_{\alpha \in A} U_\alpha = M$


 * $(2): \quad \phi_\alpha \circ \phi_\beta^{-1}$ is of class $C^k$ as a map $\phi_\beta\left(U_\alpha \cap U_\beta\right) \to \phi_\alpha\left(U_\alpha \cap U_\beta\right)$ for all $\alpha,\beta \in A$


 * $(3): \quad$ If $(U,\phi)$ is a co-ordinate system such that $\phi \circ \phi_\alpha^{-1}$ and $\phi_\alpha \circ \phi^{-1}$ are $C^k$ for all $\alpha \in A$, then $(U,\phi) \in \mathscr F$.