manifold with boundary

带边流形

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

Translation evidence: Literature usage

Meaning in context

A Hausdorff, second countable space locally homeomorphic to relatively open subsets of the closed Euclidean half-space of a fixed positive dimension; the boundary consists of points corresponding to the bounding hyperplane in such charts.

具有 Hausdorff 性和第二可数性的空间，局部同胚于固定正维数欧氏闭半空间的相对开集；对应于半空间边界超平面的点组成流形的边界。

Example

The interval $[0,1]$ is a one-dimensional manifold with boundary $\{0,1\}$. Its interior points have interval charts and its endpoints have half-interval charts.

Usage and distinctions

A manifold with boundary may have empty boundary. Intrinsic boundary differs from the boundary of an embedded subset: $S^1$ has no manifold boundary although it is its own boundary in $mathbb R^2$.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Allen Hatcher — Algebraic Topology§3.3, Other Forms of Duality, manifold-with-boundary definition, printed pp. 252–253 · Checked 2026-10-14
2. Marco Gualtieri — MAT1300F, Manifolds, 2012 course notes§1.1, Definition 1.1, second-countability convention for manifolds · Checked 2026-10-14

Translation sources

1. Wang Jiajun — Peking University, 微分拓扑, Spring 2018 course page作业表，2018-04-24，内容：带边流形、横截性定理和邻域定理 · Checked 2026-10-14

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