Covering space

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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:

• [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

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TODO: add example from reference - maybe take a picture

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