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).

Every covering map is a local homeomorphism, but the converse can fail even for a surjective map. The defining neighborhood must be covered by whole disjoint sheets simultaneously, rather than only having a suitable neighborhood around each individual point of the fiber (McMullen 2026, p. 22).

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.

See also

Branched Covering Map, Cover, Covering Space, Local Homeomorphism, Simply Connected, Topological Space, Universal Cover

Portions of this entry contributed by Todd Rowland

Explore with Wolfram|Alpha

References

Hatcher, A. Algebraic Topology. Cambridge, England: Cambridge University Press, 2002.McMullen, C. Riemann Surfaces. Course notes, Apr. 28, 2026. https://abel.math.harvard.edu/~ctm/math213b/home/course/course.pdf.

Referenced on Wolfram|Alpha

Covering Map

Cite this as:

Weisstein, Eric W., with contributions by Todd Rowland. "Covering Map." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CoveringMap.html

Subject classifications