cocountable topology

余可数拓扑

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Translation evidence: Literature usage

Meaning in context

The topology on a set whose open subsets are the empty set together with subsets having at most countable complement; finite complements are allowed.

在集合上，以空集及补集至多可数的子集作为开集的拓扑，其中允许有限补集。

Example

On $\mathbb R$, $\mathbb R\setminus\mathbb Q$ is open in the cocountable topology. It is not open in the cofinite topology because its complement is infinite.

Usage and distinctions

On a countable set this topology is discrete. On an uncountable set it is T1 and any two nonempty opens intersect; it is generally finer than the cofinite topology.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Anton Petrunin — A minimalist intro to topologyChapter 2E, Advanced exercise 2.12, cocountable topology on the real set · Checked 2026-10-14

Translation sources

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

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