[[Covering]]
# Regular covering

A **regular covering** is a [[covering]] $p : \tilde{X} \twoheadrightarrow X$ for which the [[Characteristic subgroup of a covering|characteristic subgroup]] is basepoint-invariant #m/def/homotopy 
or equivalently, the characteristic subgroup[^base] is [[Normal subgroup|normal]].

[^base]: If this holds for one basepoint $\tilde{x}_{0}$ it holds for all basepoints, see [[Characteristic conjugacy class of a path-connected covering]]

## Properties

Note that many of these properties are restricted to connected and path-connected regular coverings.

- [[Deck transformation group of a regular covering as quotient]]
- [[A covering is regular iff its deck transformation group acts transitively on fibres]]
- [[Correspondence between regular coverings and orbit spaces of their deck transformation groups]]

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