> [!definition] > > Let $X$ be a [[Set|set]], and $\alg$ be an [[Atlas|atlas]] of class $C^p$. Then the [[Maximal Atlas|maximal atlas]] containing $\alg$ is the **manifold structure** on $X$. $X$ together with the maximal atlas forms a $C^p$-**manifold**. > > The default topology on $X$ is the [[Topology on Manifold|topology]] induced by the charts. > > For simplicity, if $(U, \psi)$ is a chart in the manifold structure of $X$, then we denote it as $(U, \psi) \in X$. > > If the target of each chart is isomorphic to a [[Banach Space|Banach space]] $E$, then $X$ is an $E$-**manifold**, or that $X$ is **modelled** on $E$. > [!definition] > > If $X$ is a $E$-manifold where $E = \real^n$ is the [[Euclidean Space|n-dimensional Euclidean space]], then $X$ is $n$-dimensional. Each chart $\psi: U \to \real^n$ can be identified by its $n$ coordinate functions $\bracs{\varphi_1, \cdots, \varphi_n}$. If $P \in U$, then the coordinate map can be written as $\bracs{x_1(P), \cdots, x_n(P)}$, known as the **local coordinates** on the manifold. > [!theorem] Inherited Topological Properties > > Let $M$ be an $E$-manifold, then it inherits the following topological properties from $E$: > - [[Manifolds Inherit Compactness|Locally Compact]]: If $E$ is a [[Locally Compact|locally compact]] [[Topological Vector Space|TVS]], then $M$ is also locally compact. > - If $M$ is [[Locally Compact Hausdorff Space|LCH]] and also [[Separable Topological Space|separable]], then $M$ is [$\sigma$-compact](Sigma-Compact%20LCH%20Space). > - **First Countable**: If $E$ is [[First Countable|first countable]], then $M$ is also first countable. > - [[Manifolds Inherit Path-Connectedness|Locally Path-Connected]]: If $E$ is [[Locally Convex Topological Vector Space|locally convex]] ([[Locally Path-Connected|locally path-connected]]), then $M$ is also locally path-connected. In which case, the [[Connected Component|connected components]] of $M$ coincides with its path components. > - If $M$ is also [[Separable Topological Space|separable]], then it has only countably many connected components.