retraction onto a subspace

收缩

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

Translation evidence: Literature usage

Meaning in context

For a subspace A of a topological space X, a continuous map from X to A whose restriction to A is the identity.

对拓扑空间 $X$ 的子空间 $A$，连续映射 $r:X\to A$，其在 $A$ 上的限制为恒等映射。

Example

The map $r:S^1\to\{1\}$ that sends every point to $1$ is a retraction. It cannot be the endpoint of a deformation retraction of $S^1$ onto $\{1\}$ because their fundamental groups differ.

Usage and distinctions

A retraction supplies a continuous projection fixing the subspace; it supplies no homotopy from the identity. The Chinese name 收缩 here has this topological sense, rather than metric contraction.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Allen Hatcher — Algebraic TopologyChapter 0, Homotopy and Homotopy Type, retraction definition and point example, printed p. 3 · Checked 2026-10-14

Translation sources

1. Bao Zhiqiang — Peking University topology syllabus第11周11月23日，主要概念：收缩，形变收缩，强形变收缩 · Checked 2026-10-14

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