---
tag: Y28DDJ9F79
kind: definition
name: Covering space
msc: [57M10, 55R05]
defines: 3XZRRE165N
uses: [0VGFHTM75Y, 0QT00PB5G2, 0WS9P12FQR]
sources:
  - url: https://en.wikipedia.org/wiki/Covering_space
---

## Definition

Let $E$ and $X$ be topological spaces and let $p \colon E \to X$ be a [[continuous map|0VGFHTM75Y]].
An open set $U \subseteq X$ is *evenly covered* by $p$ if $p^{-1}(U)$ is a union of pairwise disjoint open sets $V_i \subseteq E$, $i \in J$, with one more property.
For each $i$, the restriction $p|_{V_i} \colon V_i \to U$ is a [[homeomorphism|0QT00PB5G2]], where $V_i$ and $U$ carry the [[subspace topology|0WS9P12FQR]].
The map $p$ is a *covering map* if every point of $X$ has an evenly covered open neighbourhood.
Then $E$ is a *covering space* of $X$.

## Remarks

The sets $V_i$ are the *sheets* over $U$.
Many authors also require $p$ to be surjective, or $E$ to be nonempty and connected; this varies between textbooks.
Every covering map is a [[local homeomorphism|MDH71ZZ8XX]].
Example: $p \colon \R \to S^1$, $p(t) = e^{2\pi i t}$, is a covering map.
For $z = e^{2\pi i a}$, the arc $U = p((a - 1/2, a + 1/2))$ is evenly covered, with sheets $(a - 1/2 + n, a + 1/2 + n)$ for $n \in \Z$.
Non-example: the restriction of this $p$ to $(0, 2)$ is a surjective local homeomorphism but not a covering map.
For every open arc $U$ around $1$, the part of $p^{-1}(U)$ near $0$ is a half-open piece $(0, \varepsilon)$ that is not mapped onto $U$.