Covering map (topology)/Definition

Definition

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

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

In this case $E$ is a covering space of $X$.