A coordinate chart expresses the points of a small neighborhood, usually on a manifold 
, as coordinates in Euclidean
space. An example from geography is the coordinate chart given by the functions
of latitude and longitude.
This coordinate chart is not valid on the whole globe, since it does not give unique coordinates at the
north pole or south pole
(which way is east from the north pole?).
Technically, a coordinate chart is a map
 |
where 
is an open set in 
, 
is an open set in 
, and 
is the dimension of the manifold.
Often, through notational abuse, the open set 
is equated with 
, and calculations on the manifold
are done in the coordinate chart. It must then be checked whether a change of coordinates affects the result.
The map 
must be one-to-one and in
fact must be a homeomorphism. On a smooth
manifold, it must be a diffeomorphism, although
if the coordinate chart defines the smooth structure
then this is a tautology. On a complex manifold
of complex dimension 
, a coordinate chart instead takes values in an open
set 
and is a holomorphic map, with a holomorphic
map as its inverse.
If two neighborhoods 
and 
have coordinate charts 
and 
, the transition function 
is well-defined
on 
with image 
. The overlap restriction ensures
that 
takes values in the domain of 
.
