> [!definition] > > Let $X$ be a [[Set|set]]. an **atlas** of class $C^p$ ($p \ge 0$) on $X$ is a family of pairs $(U_i, \psi_i)_{i \in I}$ such that > 1. Each $U_i \subset X$, and $\bigcup_{i \in I}U_i = X$. > 2. Each $\varphi_i: U_i \to \psi_iU_i = \widehat{U_i}$ is a bijection of $U_i$ into an [[Open Set|open set]] of a [[Banach Space|Banach space]] $E_i$, and for any $i, j$, $\psi_i(U_i \cap U_j)$ is open in $E_i$. > 3. The map $\psi_j \circ \psi_i^{-1}: \psi_i(U_i \cap U_j) \to \psi_j(U_i \cap U_j)$ is a [$C^p$](Space%20of%20Continuously%20Differentiable%20Functions)-isomorphism for each $i, j$. > > where each $(U_i, \psi_i)$ is a **chart** of the atlas. If $x \in U_i$, then $(U_i, \psi_i)$ is a **chart at** $x$. There exists a unique topology where each $U_i$ is open and each $\psi_i$ is a topological isomorphism, known as the [[Topology on Manifold|topology]] induced by the atlas. > > If the context is clear, given one chart $(U, \psi)$ and $p \in X$, $\hat p = \psi(p)$. > [!definition] > > Let $X$ be a set and $(U_i, \psi_i)_{i \in I}$ be an atlas of class $C^p$. If there exists a Banach space $E$ such that $\psi_i: U_i \to E$ for all $i \in I$, then $(U_i, \psi_i)_{i \in I}$ is an $E$-atlas. > [!theorem] Atlas Facts Rundown > > - [[Connected Components Have Isomorphic Targets]] > - If $p \ge 1$ and $U_i \cap U_j \ne \emptyset$, then $E_i$ and $E_j$ are toplinearly isomorphic. > - Let $E$ be any Banach space, then the collection of all points with a chart to a space isomorphic to $E$, is both [[Open Set|open]] and [[Closed Set|closed]]. > - Each [[Connected|connected]] subset of $X$ have charts for every point such that their targets are isomorphic to each other. > - [[Compatible Atlases]] > - Two $E$-atlases are compatible if every composition of their charts is a $C^p$ isomorphism. > - Compatibility of atlases is an [[Equivalence Relation|equivalence relation]]. > - Compatible atlases induce the same final topology. > - If all the targets of an atlas are toplinearly isomorphic, then there exists a compatible $E$-atlas. > - The [[Restriction of Atlas on Open Set|restriction of an atlas to an open set]] in the final topology is an atlas on that open set. > - [[Maximal Atlas]] > - For any atlas on $X$, there is a unique maximal atlas containing it. > - The maximal atlas containing the given atlas forms the manifold structure on $X$. > - The maximal atlas containing the given atlas is the union of all atlases compatible with it.