---
tag: 3XZRRE165N
kind: concept
name: Covering space
msc: [57M10, 55R05]
specializes: [MDH71ZZ8XX]
sources:
  - url: https://en.wikipedia.org/wiki/Covering_space
same_as:
  wikidata: Q332648
  wikipedia: 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|S1XK3HXHFX]].
They give the computation of $\pi_1(S^1)$ and link topology with group theory.

## Explanations

### L1 human

Picture an infinite spiral staircase above a circle, with each step projecting straight down onto the floor.
Near each point of the circle, the part of the staircase above it is a pile of separate small pieces, one on each turn.
That is the covering $\R \to S^1$, $t \mapsto e^{2\pi i t}$.
Another example is the map $z \mapsto z^2$ from the circle to itself, which wraps the circle twice around itself.