A covering map (also called a covering or projection) is a surjectiveopen 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."