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In mathematics, specifically algebraic topology, a covering map (also covering projection) is a continuous function p {\displaystyle p} from a topological space C {\displaystyle C} to a topological space X {\displaystyle X} such that each point in X {\displaystyle X} has an open neighborhood evenly covered by p {\displaystyle p} (as shown in the image). In this case, C {\displaystyle C} is called a covering space and X {\displaystyle X} the base space of the covering projection. The definition implies that every covering map is a local homeomorphism. Covering spaces play an important role in homotopy theory, harmonic analysis, Riemannian geometry and differential topology. In Riemannian geometry for example, ramification is a generalization of the notion of covering maps. Covering spaces are also deeply intertwined with the study of homotopy groups and, in particular, the fundamental group. An important application comes from the result that, if X {\displaystyle X} is a 'sufficiently good' topological space, there is a bijection between the collection of all isomorphism classes of connected coverings of X {\displaystyle X} and the conjugacy classes of subgroups of the fundamental group of X {\displaystyle X}. Formal definition Let X {\displaystyle X} be a topological space. A covering space of X {\displaystyle X} is a topological space C {\displaystyle C} together with a continuous surjective map p : C X {\displaystyle p\colon C\to X} such that for every x X {\displaystyle x\in X}, there exists an open neighborhood U {\displaystyle U} of x {\displaystyle x}, such that p 1 ( U ) {\displaystyle p^{-1}(U)} (the pre-image of U {\displaystyle U} under p {\displaystyle p} ) is a union of disjoint open sets in C {\displaystyle C}, each of which is mapped homeomorphically onto U {\displaystyle U} by p {\displaystyle p}. Equivalently, a covering space of X {\displaystyle X} may be defined as a fiber bundle p : C X {\displaystyle p\colon C\to X} with discrete fibers. The map p {\displaystyle p} is called the covering map, the space X {\displaystyle X} is often called the base space of the covering, and the space C {\displaystyle C} is called the total space of the covering.

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