closed manifold

闭流形

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

Translation evidence: Literature usage

Meaning in context

A compact topological manifold without boundary; manifolds here are Hausdorff and second countable.

紧致且没有边界的拓扑流形；此处的流形具有 Hausdorff 性和第二可数性。

Example

The sphere $S^2$ is closed. The compact disk $D^2$ is not a closed manifold because its boundary is $S^1$.

Usage and distinctions

Closed does not merely mean closed as a subset of some ambient space. The real line is closed in itself but is not a closed manifold because it is noncompact.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Allen Hatcher — Algebraic Topology§3.3, Poincaré Duality, closed-manifold definition, printed p. 231 · Checked 2026-10-14

Translation sources

1. Bao Zhiqiang — Peking University topology syllabus第8周10月31日，主要概念：闭流形，闭曲面 · Checked 2026-10-14

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