quotient topology

商拓扑

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

Translation evidence: Literature usage

Meaning in context

For a surjection q from a topological space X onto a set Y, the topology in which U is open in Y exactly when its inverse image under q is open in X.

对满射 q:X→Yq:X\to Y，规定 U⊆YU\subseteq Y 为开集当且仅当 q−1(U)q^{-1}(U) 在 XX 中开，由此得到的拓扑。

Example

Identifying the endpoints of [0,1][0,1] yields a quotient homeomorphic to the circle.

Usage and distinctions

A continuous surjection need not give its target the quotient topology.

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Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. Anton Petrunin — A minimalist intro to topologyChapter 6F, Quotient topology, printed p. 41 · Checked 2026-10-14

Translation sources

  1. Bao Zhiqiang — Peking University topology syllabus第7周10月26日，主要概念：商拓扑 · Checked 2026-10-14
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