Coordinate chart

A coordinate chart is a homeomorphism from an open subset of a manifold to an open subset of Euclidean space. It assigns each point in its domain a tuple of real coordinates.

Topological manifold: A space locally homeomorphic to Euclidean space and satisfying specified topological conditions. A chart supplies the local Euclidean representation required by this definition.
Coordinate map: The map that sends each point in a chart domain to its coordinate tuple. It is the function that performs a chart’s assignment of coordinates.
Local coordinates: Coordinate values assigned to points in a neighborhood of a geometric space. Chart coordinates enable local expressions for geometric objects and equations.
Global coordinate system: A coordinate assignment that represents every point of a space within one coordinate domain. A chart is local and need not cover the entire manifold.
Open set: A subset of a topological space in which every point has a neighborhood contained in the subset. Chart domains and coordinate images are open sets.
Transition map: A map between coordinate images obtained by expressing the same manifold points in two charts. It converts coordinates across overlapping chart domains.
Differential geometry: The study of smooth spaces using calculus and geometric structures. Charts provide the local coordinates used to define its central constructions.
Parametrization: A map from a parameter space into a geometric object, often used to describe a surface or curve. Unlike a chart, a parametrization need not be one-to-one or cover an open subset of the manifold.
Euclidean space: The space of finite real coordinate tuples equipped with its standard geometry and topology. A chart maps manifold points into this familiar coordinate space.
Atlas (mathematics): A collection of compatible coordinate charts covering a manifold. A single chart becomes part of a global coordinate description through an atlas.

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