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Concept
Covering space
Summary
A covering of a topological space is a continuous map with special properties.
Let be a topological space. A covering of is a continuous map
such that there exists a discrete space and for every an open neighborhood , such that and is a homeomorphism for every .
Often, the notion of a covering is used for the covering space as well as for the map . The open sets are called sheets, which are uniquely determined up to a homeomorphism if is connected. For each the discrete subset is called the fiber of . The degree of a covering is the cardinality of the space . If is path-connected, then the covering is denoted as a path-connected covering.
For every topological space , there is a covering map given by , which is called the trivial covering of
The map with is a covering of the unit circle . The base of the covering is and the covering space is . For any point such that , the set is an open neighborhood of . The preimage of under is
and the sheets of the covering are for The fiber of is
Another covering of the unit circle is the map with for some For an open neighborhood of an , one has:
A map which is a local homeomorphism but not a covering of the unit circle is with . There is a sheet of an open neighborhood of , which is not mapped homeomorphically onto .
Since a covering maps each of the disjoint open sets of homeomorphically onto it is a local homeomorphism, i.e. is a continuous map and for every there exists an open neighborhood of , such that is a homeomorphism.
It follows that the covering space and the base space locally share the same properties.
If is a connected and non-orientable manifold, then there is a covering of degree , whereby is a connected and orientable manifold.
If is a connected Lie group, then there is a covering which is also a Lie group homomorphism and is a Lie group.
If is a graph, then it follows for a covering that is also a graph.
If is a connected manifold, then there is a covering , whereby is a connected and simply connected manifold.
Official source
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