> [!definition] > > Let $X$ be a [[Set|set]], $(U_i, \psi_i)_{i \in I}$ be an [[Atlas|atlas]] of class $C^p$, then there exists exists a unique *maximal* atlas of class $C^p$ containing it, known as the **[[Manifold|manifold structure]]** determined by $(U_i, \psi_i)_{i \in I}$. > > *Proof*. Let $(V_j, \varphi_j)_{j \in J}$ be the collection of all charts that are [[Compatible Atlases|compatible]] with $(U_i, \psi_i)_{i \in I}$, then $(V_j, \varphi_j)_{j \in J}$ contains $(U_i, \psi_i)_{i \in I}$. We claim that $(V_j, \varphi_j)_{j \in J}$ is an atlas. > > Since $\seqi{U}$ covers $X$, so does $\seqj{V}$. > > As each $(V_j, \varphi_j)$ is a chart, they are bijections into open sets of Banach spaces. Let $j, j' \in J$, then since each $\psi_i$ > $ > \begin{align*} > V_j \cap V_{j'} &= \bigcup_{i \in I}V_j \cap V_{j'} \cap U_i \\ > \varphi_j(V_j \cap V_{j'}) &= \bigcup_{i \in I}\underbrace{\varphi_j \circ \psi_i^{-1}}_{C^p \text{ isomorphism}} \circ \underbrace{\psi_i(V_j \cap V_{j'} \cap U_i)}_{\text{open in }E_i} > \end{align*} > $ > the set $\varphi_j$ is an [[Open Set|open set]] in $E_j$. > > Lastly, let $j, j' \in J$, then for suitable domains, $\varphi_j \circ \varphi_{j'}^{-1}$ is a bijection. Moreover, for any $x \in X$, there exists $i \in I$ such that > $ > \varphi_j \circ \varphi_{j'}^{-1} = \underbrace{\varphi_j \circ \psi_i^{-1}}_{C^p \text{ iso}} \circ \underbrace{\psi_i \circ \varphi_{j'}^{-1}}_{C^p \text{ iso}} > $ > and its inverse is [$C^p$](Space%20of%20Continuously%20Differentiable%20Functions) at $x$. > > This collection is maximal because any "missing" chart can simply be included. More specifically, if there exists another atlas containing $(U_i, \psi_i)_{i \in I}$ is also maximal, then each chart in it is compatible with $(U_i, \psi_i)_{i \in I}$, and the proposed atlas would contain it. > [!theorem] > > Let $X$ be a set $(U_i, \psi_i)_{i \in I}$ be an atlas of class $C^p$. Then the maximal atlas is the union of all atlases [[Compatible Atlases|compatible]] with it. > > *Proof*. Since the maximal atlas containing it is compatible with $(U_i, \psi_i)_{i \in I}$ (by being an atlas and satisfying the third axiom), the maximal atlas is in the above union. As every chart in the union is compatible with the $(U_i, \psi_i)_{i \in I}$, the maximal atlas contains the union.