one-point compactification

一点紧化

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

Other names / variants: Alexandroff compactification

Translation evidence: Literature usage

Meaning in context

For a noncompact locally compact Hausdorff space X, the compact Hausdorff space formed by adding infinity, whose neighborhoods have compact closed complements in X.

对非紧局部紧 Hausdorff 空间添一个无穷远点；该点的开邻域的补集在原空间中紧且闭，由此得到紧 Hausdorff 空间。

Example

The one-point compactification of $\mathbb R$ is homeomorphic to $S^1$.

Usage and distinctions

The closed-compact-complement construction exists more generally, but the stated hypotheses ensure the intended Hausdorff compactification and dense inclusion.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Anton Petrunin — A minimalist intro to topologyChapter 7F, One-point compactification, printed pp. 49–50 · Checked 2026-10-14

Translation sources

1. Bao Zhiqiang — Peking University topology syllabus第5周10月12日，主要概念：一点紧化 · Checked 2026-10-14

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