locally compact space

局部紧致

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

Translation evidence: Literature usage

Meaning in context

A topological space in which every point lies in an open subset contained in some compact subspace; no Hausdorff condition is included in this definition.

拓扑空间的每一点都属于某个开集，且该开集包含于一个紧致子空间中；本定义不附加 Hausdorff 条件。

Example

For $x\in\mathbb R$, the open interval $(x-1,x+1)$ lies in the compact interval $[x-1,x+1]$, so $\mathbb R$ is locally compact but not compact.

Usage and distinctions

In Hausdorff spaces this agrees with having a basis of neighborhoods with compact closure. Without separation hypotheses different formulations of local compactness can diverge.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. J. Peter May — Finite Topological Spaces§7, Function spaces and homotopies, Definition 7.3, printed p. 11 · Checked 2026-10-14

Translation sources

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

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