Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, more specifically algebraic topology, a covering map is a continuous surjective function p a[¿] from a topological space, C, to a topological space, X, such that each point in X has a neighbourhood evenly covered by p. This means that for each point x in X, there is associated an ordered pair, (K, U), where U is a neighborhood of x and where K is a collection of disjoint open sets in C, each of which gets mapped homeomorphically, via p, to U (as shown in the image). In particular, this means that every covering map is necessarily a local homeomorphism. Under this definition, C is called the covering space of X. Covering spaces also play an important role in homotopy theory, harmonic analysis, Riemannian geometry and differential topology. For example: In Riemannian geometry, ramification is a generalization of the notion of covering maps. As a further example: Covering spaces are deeply interwined with the study of homotopy groups and, in particular, the fundamental group.