covering space

覆盖空间

Geometry & Topology · Undergraduate · Mathematics Resources editorial compilation (AI-assisted)

Other names / variants: 复迭空间

Translation evidence: Literature usage

Meaning in context

A space E with a continuous surjection to B such that each point of B has a neighborhood whose inverse image splits into disjoint open sheets each homeomorphic to that neighborhood.

空间 $E$ 连同连续满射 $p:E\to B$；底空间每一点都有开邻域，其原像为不交开片之并，每片由 $p$ 同胚地映到该邻域。

Example

$t\mapsto e^{2\pi it}$ covers $S^1$ by $\mathbb R$.

Usage and distinctions

A local homeomorphism is not automatically a covering map; the whole neighborhood must be evenly covered.

Read the encyclopedia explanation

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Allen Hatcher — Algebraic Topology§1.3, covering definition, printed p. 56 · Checked 2026-10-14

Translation sources

1. Li Xiaobin — Southwest Jiaotong University, Gromov–Witten study guide§1.2 基础拓扑学，核心内容：覆盖空间 · Checked 2026-10-14
2. Bao Zhiqiang — Peking University topology syllabus第13周12月7日：复迭空间；主要概念：复迭空间 · Checked 2026-10-14

Original encyclopedia and terminology content: CC BY-SA 4.0. Contribute or report a correction.