Covering space
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[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
