topological manifold

拓扑流形

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

Translation evidence: Literature usage

Meaning in context

For a fixed nonnegative integer n, a Hausdorff, second countable space in which each point has a neighborhood homeomorphic to an open subset of real n-dimensional Euclidean space; boundary is excluded in this record.

对固定的非负整数 $n$，具有 Hausdorff 性和第二可数性的空间，每一点都有同胚于 $n$ 维实欧氏空间开集的邻域；本条不含边界。

Example

The product $S^1\times S^1$ is a two-dimensional topological manifold: products of small open arcs give coordinate neighborhoods homeomorphic to open rectangles.

Usage and distinctions

No differentiability condition is imposed on chart transitions. Some authors omit second countability, yielding a broader convention; this entry includes it.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Marco Gualtieri — MAT1300F, Manifolds, 2012 course notes§1.1 Topological manifolds, Definition 1.1 and Example 1.4 · Checked 2026-10-14

Translation sources

1. Bao Zhiqiang — Peking University topology syllabus第3周9月28日，主要概念：拓扑流形；第8周10月31日：拓扑流形 · Checked 2026-10-14

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