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."