Covering Map

A covering map (also called a covering or projection) is a surjective open map f:X->Y that is locally a homeomorphism, meaning that each point in X has a neighborhood that is the same after mapping f in Y. In a covering map, the preimages f^(-1)(y) are a discrete set of X, and the cardinal number of f^(-1)(y) (which is possibly infinite) is independent of the choice of y in Y.

For example, the map f(z)=z^2, as a map f:C-0->C-0, is a covering map in which f^(-1)(y) always consists of two points. pi:C->C/Gamma=T, where Gamma={(a+bI)|a,b in Z} is another example of a covering map, and is actually the universal cover of the torus T. If f:X->T is any covering of the torus, then there exists a covering pi^~:C->X such that pi factors through pi^~, i.e., pi=f degreespi^~.

In contrast, f(z)=z^2 as a map f:C->C (with the point z=0 included) is not a true covering map, but rather a "branched covering."

See also

Cover, Covering Space, Simply Connected, Topological Space, Universal Cover

This entry contributed by Todd Rowland

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Rowland, Todd. "Covering Map." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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