Difference between revisions of "Covering space"
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| + | #REDIRECT [[Topological covering space]] | ||
| + | {{Definition|Topology|Algebraic Topology|Homotopy Theory}} | ||
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| + | {{Stub page|grade=A**|msg=What to do with the old content? | ||
| + | First chance since 15th April 2015! [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 01:26, 26 February 2017 (UTC)}} | ||
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| + | =OLD PAGE= | ||
{{Extra Maths}} | {{Extra Maths}} | ||
==Definition== | ==Definition== | ||
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==Immediate results== | ==Immediate results== | ||
| − | * The covering map is a [[ | + | * The covering map is a [[Surjection|surjection]] (it is clearly onto, as for all points in {{M|X}} - something must map to it!) |
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==Examples== | ==Examples== | ||
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Contents
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\newcommand{\bigudot}{ \mathchoice{\mathop{\bigcup\mkern-15mu\cdot\mkern8mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}} }\newcommand{\udot}{\cup\mkern-12.5mu\cdot\mkern6.25mu\!}\require{AMScd}\newcommand{\d}[1][]{\mathrm{d}^{#1} }
Definition
Here (E,\mathcal{K}) and (X,\mathcal{J}) are topological spaces
Covering projection
A map p:(E,\mathcal{K})\rightarrow(X,\mathcal{J}) is a covering projection (also known as covering map) if[1]:
- \forall x\in X\exists U\in\mathcal{J}\ \exists a non-empty collection of disjoint open sets V_\alpha such that p^{-1}(U)=\bigudot_{\alpha\in I}V_\alpha where \forall\alpha\in I we have p|_{V_\alpha}:V_\alpha\rightarrow X being a homeomorphism
Terminology
- X is the Base space of the covering map (or projection)
- E is the Covering space of the covering map (or projection)
Immediate results
- The covering map is a surjection (it is clearly onto, as for all points in X - something must map to it!)
Examples
TODO: add example from reference - maybe take a picture
