Definition:Chart
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Definition
Let $M$ be a topological manifold of dimension $d$.
A $d$-dimensional chart of $M$ is an ordered pair $\struct {U, \phi}$, where:
- $U$ is an open subset of $M$
- $\phi: U \to D$ is a homeomorphism of $U$ onto an open subset $D$ of Euclidean space $\R^d$.
Also defined as
The open set $U$ is sometimes required to be connected.
Also known as
Some sources refer to a chart as:
- a coordinate chart
- a coordinate system or local coordinate system
- a coordinate neighborhood.
The danger of confusion with other notions of coordinate renders the some of the above inadequate for use on $\mathsf{Pr} \infty \mathsf{fWiki}$.
Also see
- Results about charts can be found here.
Sources
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): chart: 2.
- 2003: John M. Lee: Introduction to Smooth Manifolds: $\S 1.1$: Smooth Manifolds. Topological Manifolds
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): chart