---
tag: Y1YPE6PM52
kind: concept
name: Universal cover
msc: [57M10, 55R05]
specializes: [3XZRRE165N]
sources:
  - url: https://en.wikipedia.org/wiki/Covering_space
---

## Summary

A universal cover of a connected space $X$ is a [[covering space|3XZRRE165N]] of $X$ whose total space is simply connected.
It exists for every connected, locally path-connected and [[semilocally simply connected|R3SHZX0NZR]] space, and it is unique up to isomorphism.
It covers every other connected covering space of $X$, and its group of deck transformations is isomorphic to the [[fundamental group|S1XK3HXHFX]] of $X$.

## Explanations

### L1 human

The universal cover unwraps every loop of a space that cannot be shrunk.
The real line wound around the circle as a helix is the universal cover of the circle.
Going once around the circle moves one step up the helix, so no loop survives upstairs.
The plane is the universal cover of the torus, through the map that reduces both coordinates modulo $1$.
The universal cover is the largest connected covering, and every other connected covering sits between it and the space.