open map

开映射

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

Translation evidence: Literature usage

Meaning in context

A map between topological spaces that sends every open subset of its domain to an open subset of its codomain; continuity and surjectivity are not included in the definition.

拓扑空间之间的映射，将定义域的每个开集映为目标空间的开集；定义本身不要求连续或满射。

Example

The projection R2→R\mathbb R^2\to\mathbb R onto the first coordinate is open. Its restriction to the closed set {(x,y):xy=1}\{(x,y):xy=1\} has image R∖{0}\mathbb R\setminus\{0\}, so the full projection is not closed.

Usage and distinctions

Continuity concerns inverse images of opens, while openness concerns direct images. An open continuous surjection is quotient, but openness alone does not assert continuity.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. Anton Petrunin — A minimalist intro to topologyChapter 5B, Closed and open maps, Definition 5.9; Chapter 6A, Exercise 6.6 · Checked 2026-10-14

Translation sources

  1. Bao Zhiqiang — Peking University topology syllabus第2周9月19日，主要概念：开/闭映射 (open-map sense) · Checked 2026-10-14
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