nLab coordinate system

Idea

A topological manifold $X$ of dimension $n \in \mathbb{N}$ is by definition a topological space that is locally homeomorphic to a Cartesian space $\mathbb{R}^n$. A choice of such morphism

$$\phi : \mathbb{R}^n \stackrel{\simeq}{\to} X|_{im(\phi)} \hookrightarrow X$$

is a coordinate system or coordinate chart or just chart on the image of $\phi$.

This generalises to other sorts of manifolds.

An atlas is the collection of coordinate charts defining a manifold structure.

graphics grabbed from Frankel

Related concepts

• coordinate transformation, diffeomorphism

• Riemann normal coordinates, Walker coordinates

• polar coordinates