Difference between revisions of "Covering space"
From Maths
m (→Immediate results) |
(Redirecting to new page!) |
||
Line 1: | Line 1: | ||
+ | #REDIRECT [[Topological covering space]] | ||
+ | {{Definition|Topology|Algebraic Topology|Homotopy Theory}} | ||
+ | |||
+ | {{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)}} | ||
+ | |||
+ | =OLD PAGE= | ||
{{Extra Maths}} | {{Extra Maths}} | ||
==Definition== | ==Definition== |
Latest revision as of 01:26, 26 February 2017
Redirect to:
Stub grade: A**
This page is a stub
This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
Contents
OLD PAGE
[math]\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}} }[/math][math]\newcommand{\udot}{\cup\mkern-12.5mu\cdot\mkern6.25mu\!}[/math][math]\require{AMScd}\newcommand{\d}[1][]{\mathrm{d}^{#1} }[/math]
Definition
Here [ilmath](E,\mathcal{K})[/ilmath] and [ilmath](X,\mathcal{J})[/ilmath] are topological spaces
Covering projection
A map [math]p:(E,\mathcal{K})\rightarrow(X,\mathcal{J})[/math] is a covering projection (also known as covering map) if[1]:
- [math]\forall x\in X\exists U\in\mathcal{J}\ \exists[/math] a non-empty collection of disjoint open sets [ilmath]V_\alpha[/ilmath] such that [math]p^{-1}(U)=\bigudot_{\alpha\in I}V_\alpha[/math] where [math]\forall\alpha\in I[/math] we have [math]p|_{V_\alpha}:V_\alpha\rightarrow X[/math] being a homeomorphism
Terminology
- [ilmath]X[/ilmath] is the Base space of the covering map (or projection)
- [ilmath]E[/ilmath] 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 [ilmath]X[/ilmath] - something must map to it!)
Examples
TODO: add example from reference - maybe take a picture