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On pg. 4 of Guillemin and Pollack, after defining $k$-dimensional manifold, the authors define a coordinate system as the diffeomorphism $\phi^{-1}:V\rightarrow U$, where $U\subset \mathbb{R}^k$, and $V$ is a neighborhood in the $k$-dimensional neighborhood $X$ ($\phi^{-1}$ is the inverse of the parametrization $\phi:U\rightarrow V$).

The next sentence is what I'm confused about. They state: "When we write the map $\phi^{-1}$ in coordinates $\phi^{-1}=(x_1,...,x_k)$, the $k$ smooth functions $x_1,...,x_k$ on V are called coordinate functions," and are sometimes called "local coordinates". Why are these smooth functions? Is $(x_1,...,x_k)$ supposed to be thought of as a point in the manifold $X$? Isn't $(x_1,...,x_k)$ a points in $U$?

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For all $i\in\{1,\ldots,k\}$, let $\pi_i\colon\mathbb{R}^k\rightarrow\mathbb{R}$ be the projection onto the $i$th factor, namely: $$\pi_i(y_1,\ldots,y_k)=y_i,$$ then $x_i\colon V\rightarrow\mathbb{R}$ is defined to be $\pi_i\circ\phi^{-1}$ so it is smooth as a composition of smooth maps.

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