closed map

闭映射

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

Translation evidence: Literature usage

Meaning in context

A map between topological spaces that sends every closed subset of its domain to a closed subset of its codomain; it need not be continuous or surjective merely by this definition.

拓扑空间之间的映射，将定义域的每个闭集映为目标空间的闭集；该定义本身不推出连续性或满射性。

Example

The inclusion [0,1]↪R[0,1]\hookrightarrow\mathbb R is closed: its domain is compact and the target Hausdorff. It is not open since [0,1/2)[0,1/2) is relatively open but not open in R\mathbb R.

Usage and distinctions

A map with a closed graph need not be a closed map. Closedness is a property of direct images, whereas continuity tests inverse images.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. Anton Petrunin — A minimalist intro to topologyChapter 5B, Definition 5.9; Chapter 9D, Observation 9.15 · 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 (closed-map sense) · Checked 2026-10-14
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