Covering space - Revision history

Karl Jones: /* See also */

2016-09-19T13:45:59Z

‎See also

Karl Jones: /* See also */

2016-09-19T13:44:54Z

‎See also

Karl Jones: /* External links */

2016-04-22T21:19:11Z

‎External links

Karl Jones: /* See also */

2016-02-07T10:13:55Z

‎See also

Karl Jones at 16:26, 5 February 2016

2016-02-05T16:26:09Z

Karl Jones: /* Description */

2015-09-14T10:32:49Z

‎Description

Karl Jones: Created page with "In mathematics, more specifically algebraic topology, a '''covering map''' (also '''covering projection''') is a continuous Function (mathema..."

2015-09-14T10:32:22Z

Created page with "In mathematics, more specifically algebraic topology, a '''covering map''' (also '''covering projection''') is a continuous Function (mathema..."

New page

In [[mathematics]], more specifically [[algebraic topology]], a '''covering map''' (also '''covering projection''') is a [[continuous function|continuous]] [[Function (mathematics)|function]] ''p'' from a [[topological space]], ''C'', to a topological space, ''X'', such that each point in ''X'' has an [[open set|open]] neighbourhood '''evenly covered''' by ''p'' (as shown in the image)

In this case, ''C'' is called a '''covering space''' and ''X'' the '''base space''' of the covering projection.

(TO DO: expand, organize, cross-reference, illustrate.)

== Description ==

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 (mathematics)#In algebraic topology|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'' is a "sufficiently good" [[topological space]], there is a [[bijection]] between the collection of all isomorphism classes of [[connected space|connected coverings]] of ''X'' and the subgroups of the [[fundamental group]] of ''X''.

== See also ==

* [[Mathematics]]

== External links ==

* [https://en.wikipedia.org/wiki/Covering_space Covering space] @ Wikipedia

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 * [[Covering group]]
 * [[Degrees of freedom]]
 * [[Galois connection]]

← Older revision		Revision as of 21:19, 22 April 2016
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	* [https://en.wikipedia.org/wiki/Covering_space Covering space] @ Wikipedia		* [https://en.wikipedia.org/wiki/Covering_space Covering space] @ Wikipedia
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		+	[[Category:Mathematics]]

← Older revision		Revision as of 10:13, 7 February 2016
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	== See also ==		== See also ==

		+	* [[Degrees of freedom]]
	* [[Mathematics]]		* [[Mathematics]]

← Older revision		Revision as of 16:26, 5 February 2016
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	In this case, ''C'' is called a '''covering space''' and ''X'' the '''base space''' of the covering projection.		In this case, ''C'' is called a '''covering space''' and ''X'' the '''base space''' of the covering projection.
−
−	~~(TO DO: expand, organize, cross-reference, illustrate.)~~

	== Description ==		== Description ==

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 == See also ==
 * [[Degrees of freedom]]
-* [[Mathematics]]
+* [[Galois connection]]
 == External links ==

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 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 (mathematics)#In algebraic topology|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'' is a "sufficiently good" [[topological space]], there is a [[bijection]] between the collection of all isomorphism classes of [[connected space|connected coverings]] of ''X'' and the subgroups of the [[fundamental group]] of ''X''.
+Covering spaces play an important role in [[homotopy theory]], [[harmonic analysis]], [[Riemannian geometry]] and [[differential topology]]. In Riemannian geometry for example, [[Ramification (mathematics)#In algebraic topology|ramification]] is a generalization of the notion of covering maps.
 == See also ==