[[Manifold]]
# Coördinate chart

Let $M$ be a [[topological space]].
A **coördinate chart** $(x,U)$ is an embedding of an open subset $U \sube M$ in $\mathbb{R}^n$
$$
\begin{align*}
x : U &\hookrightarrow \mathbb{R}^n \\
p &\mapsto (x^\mu(p))
\end{align*}
$$
In a [[topological manifold]], the most basic kind of manifold, every point has a [[Topological manifold|Euclidean neighbourhood]] $U$,
and the coördinate chart essentially translates between a manifold and the section of Euclidean space it resembles.
The inverse of a coördinate chart is called a **local paramaterization**,
and setting $\varphi^{-1}(0) = p$ gives a local parameterization at $p$.

## See also
- A collection of charts covering a whole manifold is called an [[atlas]].

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