universal covering space
Let be a topological space. A universal covering space is a covering space of which is connected and simply connected.
If is based, with basepoint , then a based cover of is cover of which is also a based space with a basepoint such that the covering is a map of based spaces. Note that any cover can be made into a based cover by choosing a basepoint from the pre-images of .
The universal covering space has the following universal property: If is a based universal cover, then for any connected based cover , there is a unique covering map such that .
Clearly, if a universal covering exists, it is unique up to unique isomorphism. But not every topological space has a universal cover. In fact has a universal cover if and only if it is semi-locally simply connected (for example, if it is a locally finite CW-complex or a manifold).
Title | universal covering space |
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Canonical name | UniversalCoveringSpace |
Date of creation | 2013-03-22 12:15:34 |
Last modified on | 2013-03-22 12:15:34 |
Owner | bwebste (988) |
Last modified by | bwebste (988) |
Numerical id | 7 |
Author | bwebste (988) |
Entry type | Definition |
Classification | msc 54-00 |
Synonym | universal cover |
Related topic | OmegaSpectrum |