cofinite topology

余有限拓扑

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

Translation evidence: Literature usage

Meaning in context

The topology on a set whose open subsets are the empty set together with subsets having finite complement.

在集合上，以空集及补集为有限集的子集作为开集的拓扑。

Example

On $\mathbb N$, $\mathbb N\setminus\{1,4\}$ is open. Any two nonempty opens intersect because the union of their finite complements cannot exhaust $\mathbb N$.

Usage and distinctions

On an infinite set this topology is T1 but not Hausdorff. On a finite set it coincides with the discrete topology, so the infinite-set qualification matters.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Anton Petrunin — A minimalist intro to topologyChapter 2B, Examples, cofinite topology and Exercise 2.2 · Checked 2026-10-14

Translation sources

1. Bao Zhiqiang — Peking University topology syllabus第1周9月14日，常用例子：余有限拓扑 · Checked 2026-10-14

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