Covering map (topology)/Definition

Definition

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 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].