Intuitively, a covering locally projects a "stack of pancakes" above an open neighborhood onto
In topology, a covering or covering projection is a map between topological spaces that, intuitively, locally acts like a projection of multiple copies of a space onto itself. In particular, coverings are special types of local homeomorphisms. If is a covering, is said to be a covering space or cover of , and is said to be the base of the covering, or simply the base. By abuse of terminology, and may sometimes be called covering spaces as well. Since coverings are local homeomorphisms, a covering space is a special kind of étale space.
Covering spaces first arose in the context of complex analysis (specifically, the technique of analytic continuation), where they were introduced by Riemann as domains on which naturally multivalued complex functions become single-valued. These spaces are now called Riemann surfaces.[1]: 10
Covering spaces are an important tool in several areas of mathematics. In modern geometry, covering spaces (or branched coverings, which have slightly weaker conditions) are used in the construction of manifolds, orbifolds, and the morphisms between them. In algebraic topology, covering spaces are closely related to the fundamental group: for one, since all coverings have the homotopy lifting property, covering spaces are an important tool in the calculation of homotopy groups. A standard example in this vein is the calculation of the fundamental group of the circle by means of the covering of by (see below).[2]: 29 Under certain conditions, covering spaces also exhibit a Galois correspondence with the subgroups of the fundamental group.
^Forster, Otto (1981). "Chapter 1: Covering Spaces". Lectures on Riemann Surfaces. GTM. Translated by Bruce Gillian. New York: Springer. ISBN 9781461259633.
^Hatcher, Allen (2001). Algebraic Topology. Cambridge: Cambridge Univ. Press. ISBN 0-521-79160-X.
covering or covering projection is a map between topological spaces that, intuitively, locally acts like a projection of multiple copies of a space onto...
{\displaystyle X} Coveringspace, a certain kind of continuous maps Covering (martial arts), an act of protecting against an opponent's strikes The Covering, a studio...
and the pia mater. Cerebrospinal fluid is located in the subarachnoid space between the arachnoid mater and the pia mater. The primary function of the...
exhibits the portion y ≥ 1 of the upper half-plane as the universal coveringspace of the pseudosphere. The precise mapping is ( x , y ) ↦ ( v ( arcosh...
seen directly without using coveringspaces. The group G is called the universal covering group of H. As the universal covering group suggests, there is...
mathematics, a covering group of a topological group H is a coveringspace G of H such that G is a topological group and the covering map p : G → H is...
bundles include the Möbius strip and Klein bottle, as well as nontrivial coveringspaces. Fiber bundles, such as the tangent bundle of a manifold and other...
action on Sn. As mentioned above, the orbit space for this action is RPn. This action is actually a coveringspace action giving Sn as a double cover of RPn...
{\displaystyle \chi (E)=\chi (F)\cdot \chi (B).} This includes product spaces and coveringspaces as special cases, and can be proven by the Serre spectral sequence...
Lebesgue covering dimension or topological dimension of a topological space is one of several different ways of defining the dimension of the space in a topologically...
topology and the projection sending (x, o) to x is then a 2-to-1 covering map. This coveringspace is called the orientable double cover, as it is orientable...
^{1}} which ramifies at three points. Then, there is an associated coveringspace C | U → U = P 1 − { p 1 , p 2 , p 3 } {\displaystyle C|_{U}\to U=\mathbb...
connected based topological space with base point x, and let p : X ~ → X {\displaystyle p:{\tilde {X}}\to X} be a covering with fiber F = p − 1 ( x ) {\displaystyle...
but examples of coveringspaces of topological spaces, so the terminology in the theory of coveringspaces is available; say covering transformation group...
several loosely related functors that generalise the functors taking a coveringspace π:X→S{\displaystyle \pi \colon X\rightarrow S} to the fiber π−1(s){\displaystyle...
a Riemannian metric. More generally, but by the same principle, any coveringspace of a Riemannian manifold inherits a Riemannian metric. Also, an immersed...
In mathematics, a covering number is the number of balls of a given size needed to completely cover a given space, with possible overlaps between the...
lifting property or the covering homotopy axiom) is a technical condition on a continuous function from a topological space E to another one, B. It is...
bi-monthly educational publication coveringspace tourism and space exploration developments in companies like SpaceX, Orbital Sciences, Virgin Galactic...
Euler characteristic of a space and a ramified cover. For example, hyperbolic Riemann surfaces are ramified coveringspaces of the sphere (they have non-constant...
universal coveringspace, though not all authors include the assumption on the homotopy type. For example, any topological group is a simple space (provided...
}}\gamma (1)=x_{0}\right\}} which is the loop space Ω X {\displaystyle \Omega X} . Given a universal covering π : X ~ → X {\displaystyle \pi :{\tilde {X}}\to...
mathematics, the Vitali covering lemma is a combinatorial and geometric result commonly used in measure theory of Euclidean spaces. This lemma is an intermediate...
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