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 element 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. π:C->C/Γ≃T, where Γ = {(a + b I)|a, b element 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 π^~ :C->X such that π factors through π^~, i.e., π = f°π^~.

cover | covering space | simply connected | topological space | universal cover