A covering map (also called a covering or projection) is a continuous surjection 
such that every point 
has an open set 
containing it for which 
is a disjoint union
of open sets, each mapped homeomorphically onto
by 
. The preimage 
is a discrete set.
Its cardinal number, possibly infinite, is constant
on each connected component 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 map.
