Editing Covering space (section)

=== Definition ===
Let <math>p:E \rightarrow X</math> be a covering. A '''deck transformation''' is a homeomorphism <math>d:E \rightarrow E</math>, such that the diagram of continuous maps
[[File:Diagramm_Decktrafo.png|center|frameless]]
commutes. Together with the composition of maps, the set of deck transformation forms a [[Group (mathematics)|group]] <math>\operatorname{Deck}(p)</math>, which is the same as <math>\operatorname{Aut}(p)</math>.

Now suppose <math>p:C \to X</math> is a covering map and <math>C</math> (and therefore also <math>X</math>) is connected and locally path connected. The action of <math>\operatorname{Aut}(p)</math> on each fiber is [[Group action (mathematics)#Notable properties of actions|free]]. If this action is [[Group action (mathematics)#Remarkable properties of actions|transitive]] on some fiber, then it is transitive on all fibers, and we call the cover '''regular''' (or '''normal''' or '''Galois'''). Every such regular cover is a [[principal bundle|principal {{nowrap|<math>G</math>-bundle}}]], where <math>G = \operatorname{Aut}(p)</math> is considered as a discrete topological group.

Every universal cover <math>p:D \to X </math> is regular, with deck transformation group being isomorphic to the [[fundamental group]] {{nowrap|<math>\pi_1(X)</math>.}}