Definition

A cover is a continuous map from topological spaces $E$ and $X$

$p : E ⟶ X$

such that:

$p$ is surjective
For each $x \in X$ , there exists an open set $x \in U \subset X$ such that each path component of $p^{- 1} (U) \subset E$ is mapped homeomorphically onto $U$ .

Terminology

$X$ - base

$E$ - total space

$p^{- 1} (x)$ - fiber of $p$ over $x$

$U$ - fundamental neighborhood of $x$

For covers $p : (E_{1}, e_{1}) \to (X, x_{0})$ and $p_{2} : (E_{2}, e_{2}) \to (X, x_{0})$ , a morphism of covers is a continuous function

$f : E_{1} ⟶ E_{2}$ such that $f (e_{1}) = e_{2}$ and $p_{2} \circ f = p_{1}$ .

Thus, the diagram below commutes

A special type of morphism is from a cover $E$ back to itself. Thus, the diagram from above would become

We can denote the set of all automorphisms as $Aut_{X} (E) or Aut_{X} E .$

Let $f_{1}, f_{2} : E_{1} \to E_{2}$ be morphisms of covers. If $f_{1} (e) = f_{2} (e)$ for some $e \in E$ , then $f_{1} = f_{2}$

Proof

todo (On one of my algebraic topology HW’s)

Example 1: Trivial covering

$i d : X ⟶ X$

Example 2:

R t ⟶ S^{1} ⟼ (cos t, sin t)

Example 3:

S^{1} (cos t, sin t) ⟶ S^{1} ⟼ (cos n t, sin n t)