---- > [!definition] Definition. ([[coordinate patch]]) > A **coordinate patch** $\alpha: \mathcal{U} \to V$ on a Let $M$ be an $n$-[[topological manifold]]. A **coordinate patch on $M$** is a pair $(\widehat{U}, \alpha)$, where $\widehat{U}$ is an open subset of $\mathbb{R}^{n}$ and $\alpha:\widehat{U} \to U$ is a [[homeomorphism]] from $\widehat{U}$ to an [[subspace topology|open subset]] $\widehat{U}=\alpha(U) \subset \mathbb{R}^{n}$. > \ > Note that $\alpha ^{-1}$ a [[coordinate chart]], often denoted $\varphi: U \to \widehat{U}$. > > \ > ![[CleanShot 2023-01-10 at 18.13.32.jpg]] > ---- #### ---- #### 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 > ```