An atlas is a collection of consistent coordinate charts whose domains cover a manifold, where "consistent" most commonly means that the transition functions of the coordinate charts are smooth. As the name suggests, an atlas corresponds to a collection of maps, each of which shows a piece of a manifold and looks like flat Euclidean space. To use an atlas, one needs to know how the maps overlap. To be useful, the maps must not be too different on these overlapping areas.
The overlapping maps from one coordinate chart to another are called transition functions.
They represent the transition from one coordinate
chart's point of view to that of another. Let the unit open
ball in 
be denoted 
. Then if 
and 
are two coordinate
charts, the composition 
is a function
defined on 
. That is, it is
a function from an open subset of 
to 
,
and given such a function from 
to 
,
there are conditions for it to be smooth or have
continuous derivatives (i.e., it is a Ck
function). Furthermore, when 
is isomorphic to 
(in the even dimensional
case), a function can be holomorphic.
A smooth atlas has transition functions that are C∞ smooth
(i.e., infinitely differentiable). The consequence
is that a smooth function on one coordinate
chart is smooth in any other coordinate
chart (by the chain rule for higher derivatives).
Similarly, one could have an atlas in class 
, where the transition
functions are in class Ck.
In the even-dimensional case, one may ask whether the transition functions are holomorphic. In this case, one has a holomorphic atlas, and by the chain rule, it makes sense to ask if a function on the manifold is holomorphic.
It is possible for two atlases to be compatible, meaning the union is also an atlas. By Zorn's lemma, there always exists a maximal atlas, where a maximal atlas is an atlas not contained in any other atlas. However, in typical applications, it is not necessary to use a maximal atlas and any sufficiently refined atlas will do.
