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
(Hatcher 2002, p. 56).
For example, the map 
from 
to 
is a covering map in which 
always consists of two points.
The map 
, where 
, is another covering map and is the universal cover of the torus 
. If 
is a covering map with 
connected, there exists
a covering map 
such that 
(Hatcher 2002, pp. 61-62).
In contrast, the map 
from 
to 
,
with the point 
included, is a branched
covering map rather than a covering map.
