Covering Map

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."