# Topological covering space

(Redirected from Covering space (topology))
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## Definition

Let [ilmath](X,\mathcal{ J })[/ilmath] be a topological space. We say "[ilmath]X[/ilmath] is covered by [ilmath]E[/ilmath]" or "[ilmath]E[/ilmath] is a covering space for [ilmath]X[/ilmath]" if[1]:

1. [ilmath](E,\mathcal{ H })[/ilmath] is a topological space itself; and
2. there exists a covering mapor: bellow of the form: [ilmath]p:E\rightarrow X[/ilmath]

### Covering map

Let [ilmath](X,\mathcal{ J })[/ilmath] and [ilmath](E,\mathcal{ H })[/ilmath] be topological spaces. A map, [ilmath]p:E\rightarrow X[/ilmath] between them is called a covering map[1] if:

1. [ilmath]\forall U\in\mathcal{J}[p^{-1}(U)\in\mathcal{H}][/ilmath] - in words: that [ilmath]p[/ilmath] is continuous
2. [ilmath]\forall x\in X\exists e\in E[p(e)\eq x][/ilmath] - in words: that [ilmath]p[/ilmath] is surjective
3. [ilmath]\forall x\in X\exists U\in\mathcal{O}(x,X)[U\text{ is } [/ilmath][ilmath]\text{evenly covered} [/ilmath][ilmath]\text{ by }p][/ilmath] - in words: for all points there is an open neighbourhood, [ilmath]U[/ilmath], such that [ilmath]p[/ilmath] evenly covers [ilmath]U[/ilmath]

In this case [ilmath]E[/ilmath] is a covering space of [ilmath]X[/ilmath].