Editing Covering space (section)

{{Short description|Type of continuous map in topology}}

[[File:Covering space diagram.svg|upright=1|thumb|Intuitively, a covering locally projects a "stack of pancakes" above an [[open neighborhood]] <math>U</math> onto <math>U</math>]]

In [[topology]], a '''covering''' or '''covering projection''' is a [[continuous function|map]] between [[topological space]]s that, intuitively, [[Local property|locally]] acts like a [[Projection (mathematics)|projection]] of multiple copies of a space onto itself. In particular, coverings are special types of [[local homeomorphism]]s. If <math> p : \tilde X \to X </math> is a covering, <math>(\tilde X, p)</math> is said to be a '''covering space''' or '''cover''' of <math>X</math>, and <math>X</math> is said to be the '''base of the covering''', or simply the '''base'''. By [[abuse of terminology]], <math>\tilde X</math> and <math>p</math> may sometimes be called '''covering spaces''' as well. Since coverings are local homeomorphisms, a covering space is a special kind of [[étalé space]].

Covering spaces first arose in the context of [[complex analysis]] (specifically, the technique of [[analytic continuation]]), where they were introduced by [[Bernhard Riemann|Riemann]] as domains on which naturally [[multivalued function|multivalued]] complex functions become single-valued. These spaces are now called [[Riemann surface]]s.<ref name="Gillian">{{Cite book|translator=Bruce Gillian|last=Forster|first=Otto|title=Lectures on Riemann Surfaces|publisher=Springer|year=1981|series=GTM|number=81|chapter=Chapter 1: Covering Spaces|location=New York|ISBN=9781461259633}}</ref>{{rp|p=10}}

Covering spaces are an important tool in several areas of mathematics. In modern [[geometry]], covering spaces (or [[branched covering]]s, which have slightly weaker conditions) are used in the construction of [[manifold]]s, [[orbifold]]s, and the [[morphism]]s 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 [[Circle|<math>S^1</math>]] by [[Real number|<math>\mathbb{R}</math>]] (see [[Covering_space#Lifting_property|below]]).<ref name="Hatcher">{{Cite book|last=Hatcher|first=Allen|title=Algebraic Topology|publisher=Cambridge Univ. Press|year=2001|isbn=0-521-79160-X|location=Cambridge}}</ref>{{rp|p=29}} Under certain conditions, covering spaces also exhibit a [[Galois_connection#Algebraic_topology:_covering_spaces|Galois correspondence]] with the subgroups of the fundamental group.