Covering Map
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المرجع الالكتروني للمعلوماتيه
المصدر:
www.almerja.com
الجزء والصفحة:
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18-7-2021
1930
Covering Map
A covering map (also called a covering or projection) is a surjective open map 
that is locally a homeomorphism, meaning that each point in 
has a neighborhood that is the same after mapping 
in 
. In a covering map, the preimages 
are a discrete set of 
, and the cardinal number of 
(which is possibly infinite) is independent of the choice of 
.
For example, the map 
, as a map 
, is a covering map in which 
always consists of two points. 
, where ![Gamma=<span style=]()
{(a+bI)|a,b in Z}" src="https://mathworld.wolfram.com/images/equations/CoveringMap/Inline13.gif" style="height:18px; width:138px" /> is another example of a covering map, and is actually the universal cover of the torus 
. If 
is any covering of the torus, then there exists a covering 
such that 
factors through 
, i.e., 
.
In contrast, 
as a map 
(with the point 
included) is not a true covering map, but rather a "branched covering."
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