---- > [!definition] Definition. ([[smooth structure]]) > A **smooth structure** on a [[topological manifold]] $M$ is a maximal [[atlas of a manifold|smooth atlas]].[^1] ^definition A **smooth structure** on an arbitrary *set* $M$ (not yet with a topology) is a collection of $\{ U_{\alpha} \}$ of subsets of $M$ together with [[bijection|bijections]] $\varphi_{\alpha}: U_{\alpha} \to V_{\alpha} \overbrace{\subset}^{\text{open in }} \mathbb{R}^{n}$ such that the following properties are satisfied: 1. (Covering) Countably many $U_{\alpha}$ cover $M$ 2. (Compatibility) For all $\alpha, \beta$, $\varphi_{\beta} \circ \varphi_{\alpha} ^{-1}$ is [[continuously differentiable|smooth]] when it needs to be (i.e., on $\varphi_{\alpha}(U_{\alpha} \cap U_{\beta})$) 3. For each $\alpha$ and $\beta$, the sets $\varphi_{\alpha}(U_{\alpha} \cap U_{\beta })$ and $\varphi_{\beta}(U_{\alpha} \cap U_{\beta})$ are open in $\mathbb{R}^{n}$. 4. (Maximality) If $\varphi$ is compatible with all $\varphi_{\alpha}$, then $\varphi$ is in the collection [^1] The **induced topology on $M$** by the $\{ U_{\alpha} \}$ is that which makes the $\varphi_{\alpha}$ into [[homeomorphism|homeomorphisms]], i.e., that [[topology generated by a basis|generated by]] the [[basis for a topology|basis]] $\{ \varphi_{\alpha} ^{-1}(V_{\alpha}) \}$. > [!justification]- > - [ ] prove that the induced topology is indeed a topology and that we get a unique smooth structure from it (i think). maybe Lee 21 helps ^justification ---- #### [^1]: Here, 'maximal' means that if a chart $\varphi$ is [[transition map|smoothly compatible]] with every element of $\mathscr{A}$, then in fact $\varphi \in \mathscr{A}$. ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```