Homotopy theory MOC

Covering

Let and be topological spaces. A covering¹ of with is a surjective map such that every has a a neighbourhood such that and is a homeomorphism for each . topology A neighbourhood with these properties is called an evenly covered neighbourhood, each is called a sheet, and the discrete set is called the fibre of .²

Further terminology

• A covering is called a (path-)connected covering iff (and therefore ) is (path-)connected.
• For a connected covering, the sheet number is the same everywhere. See Sheet number of a connected covering
• The coverings of a space form a Category of coverings
• The Characteristic subgroup of a covering is its image on the fundamental group
• A Regular covering has a basepoint-invariant characteristic subgroup
• A Deck transformation is an automorphism of the covering space such that .

Properties

• The identity map is a trivial covering
• A covering is a Local homeomorphism
• The Orbit space of a properly discontinuous group action comes with a covering
• Lift of a map to a covering space
• A covering is injective on the fundamental group
• Main theorem of coverings

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Footnotes

1. German Überlagerung ↩

2. Eine Umgebung mit diesen Eigenschaften heißt eine gleichmäßig überlagerte Umgebung, jedes heißt ein Blatt, und die diskrete Menge heißt die Faser von . ↩