Coordinate space
In mathematics, a coordinate space is a space in which an ordered list of coordinates, each from a set (not necessarily the same set), collectively uniquely determine an element (or point) of the space – in short, a space with a coordinate system.
By adding further structure and constraints, a coordinate space may be used to construct an object such as a vector space or a manifold.
Definition
Let Ci, 1 ≤ i ≤ n, be n sets. A coordinate space of n dimensions is a set S together with a surjective partial mapping φ : C1 × ... × Cn → S.
Discussion
In many mathematical and practical problems, being able to index the elements of a space through the structure of an n-tuple provides one natural way in which to build the structure of an object. Examples include n-dimensional vector spaces and manifolds, geometries and fibre bundles.
Further structure may be assigned to the coordinate space through the coordinates. For example, an n-dimensional K-vector space may be defined as a coordinate space with the added structure of K-linearity in each of its n coordinates.
It is in general possible to assign a different mapping from a new set of coordinates to the same coordinate space, as, for example with a change of basis for a vector space.
It is not necessary that every n-tuple from the cartesian product of the coordinate sets maps to an element of the space, nor is it necessary that every element have a unique set of coordinates. For example, a geographic coordinate system might assign coordinates of latitude and longitude to locations on the surface of the Earth, but here it will be necessary either to constrain the allowed coordinate pairs or to allow distinct sets of coordinates for the same points.