A covering map (also called a covering or projection) is a surjective open map 
that is locally a homeomorphism,
meaning that each point in 
has a neighborhood that is the same after mapping
in 
. In a covering map, the preimages 
are a discrete
set of 
,
and the cardinal number of 
(which is possibly infinite) is independent of the
choice of 
.
For example, the map 
,
as a map 
,
is a covering map in which 
always consists of two points. 
, where 
is another example of a covering map,
and is actually the universal cover of the torus 
. If 
is any covering of the torus,
then there exists a covering 
such that 
factors through 
, i.e., 
.
In contrast, 
as a map 
(with the point 
included) is not a true covering map, but rather a "branched covering."
