compactness

紧致性

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

Other names / variants: compactness of a topological space · 紧致

Translation evidence: Literature usage

Meaning in context

The property that every open cover of a topological space admits a finite subcover; Hausdorff separation is not imposed here.

拓扑空间的每个开覆盖均有有限子覆盖的性质；此处不把 Hausdorff 条件包含在定义内。

Example

[0,1][0,1] is compact; the cover (1/n,1)(1/n,1) shows (0,1)(0,1) is not.

Usage and distinctions

Closed and bounded characterizes compact subsets of Euclidean space, not arbitrary spaces.

Read the encyclopedia explanation

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. Anton Petrunin — A minimalist intro to topologyChapter 7A, Definition, printed p. 45 · Checked 2026-10-14

Translation sources

  1. Bao Zhiqiang — Peking University topology syllabus第5周10月12日：紧致与列紧；特别关注：紧致性 · Checked 2026-10-14
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