# Topological covering space

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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]}:

- [ilmath](E,\mathcal{ H })[/ilmath] is a topological space itself; and
- there exists a covering map
^{or: 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:

- [ilmath]\forall U\in\mathcal{J}[p^{-1}(U)\in\mathcal{H}][/ilmath] - in words: that [ilmath]p[/ilmath] is continuous
- [ilmath]\forall x\in X\exists e\in E[p(e)\eq x][/ilmath] - in words: that [ilmath]p[/ilmath] is surjective
- [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].

## See next

## See also

- Covering map
- may have some useful information on it! TODO: Pertaining to properties of the covering map itself- should they be reproduced here? Alec (talk) 01:35, 26 February 2017 (UTC)
- Evenly covered

- may have some useful information on it!