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Let $X$ be a Topological Space. A Coordinate Chart of Manifold Dimension $n$ on $X$ is a pair ($U$, $\varphi$) where

1. $U \subset X$ is an Open set
2. $\varphi : U \to \hat{U}$ is a Homeomorphism for some $\hat{U} \subset \mathbb{R}^{n}$ Open.

1. For a Coordinate Chart $(U, \varphi)$, we will often use the phrase that this Coordinate Chart “contains” $p \in M$. This just means $p \in U$.
2. For some Open $\hat{V} \subset \mathbb{R}^{n}$, if we say that $(U, \varphi)$ is a Coordinate Chart to $\hat{V}$, we mean that $\varphi$ has Codomain $\hat{V}$.