Covering space

Builds on Continuous map Homeomorphism Subspace topology

Definition

Let $E$ and $X$ be topological spaces and let $p \colon E \to X$ be a continuous map. 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, where $V_i$ and $U$ carry the subspace topology. 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. 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 $U$ .

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Continuous map concept
Homeomorphism concept
Subspace topology concept
Covering space definition

Tag	`Y28DDJ9F79`
URI	`m2://Y28DDJ9F79`
Kind	definition
Writing	W0 stub
MSC	57M10, 55R05
Defines	Covering space
Uses	Continuous map, Homeomorphism, Subspace topology
Changed	2 November 2026

Covering space

Definition

Remarks

Reading path

Further reading

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