neighborhood basis

邻域基

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

Other names / variants: local base

Translation evidence: Literature usage

Meaning in context

At a point of a topological space, a family of neighborhoods such that every neighborhood of that point contains a member of the family; here a neighborhood may be any set containing an open set that contains the point.

在拓扑空间的一点处，由该点邻域组成的族，使任意邻域都包含族中某个成员；此处邻域允许是含有该点开邻域的任意集合。

Example

The intervals $(-1/n,1/n)$, for positive integers $n$, form a neighborhood basis at zero in the usual real line.

Usage and distinctions

Members need not be open under the stated convention, unlike the separate open-neighborhood sense. A neighborhood basis belongs to one point; a topology basis describes opens throughout the space.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Michael Fochler — Math 330 - Additional Material§7.1.4, Definition 7.12, Basis and neighborhood basis · Checked 2026-10-14

Translation sources

1. Bao Zhiqiang — Peking University topology syllabus第2周9月26日，主要概念及特别关注：邻域基 · Checked 2026-10-14

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