Covering space

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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:

• $$\forall x\in X\exists U\in\mathcal{J}\ \exists$$ 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

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TODO: add example from reference - maybe take a picture

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