Definition

A cover is a continuous map from topological spaces and

such that:

is surjective
For each , there exists an open set such that each path component of is mapped homeomorphically onto .

Terminology

- base

- total space

- fiber of over

- fundamental neighborhood of

For covers and , a morphism of covers is a continuous function

such that and .

Thus, the diagram below commutes

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

We can denote the set of all automorphisms as

Let be morphisms of covers. If for some , then

Proof

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

Example 1: Trivial covering

Example 2:

Example 3: