Topological covering space

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Be sure to contrast with other books! There is variation here, I need to know how it varies!

Definition

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

1. $(E,\mathcal{ H })$ is a topological space itself; and
2. there exists a covering map^{or: bellow} of the form: $p:E\rightarrow X$

Covering map

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 $X$.

See next

• Lifting of a continuous map through a covering map

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