Covering space

Summary

A covering space of a space $X$ is a space $E$ with a continuous map $p \colon E \to X$ of a special kind. Over a small neighbourhood of each point of $X$, the map looks like a stack of disjoint copies of that neighbourhood. Such a map $p$ is a covering map. Covering spaces have the path lifting and homotopy lifting properties, and for nice spaces they are classified by subgroups of the fundamental group. They give the computation of $\pi_1(S^1)$ and link topology with group theory.