LemmaStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02

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A locally finite open cover by subspaces with $\sigma$ -locally-finite bases yields a $\sigma$ -locally-finite basis of the whole space

Statement

Let $\mathcal U$ be a locally finite open cover of $X$ . If every $U\in\mathcal U$ , with its subspace topology, has a $\sigma$ -locally-finite open basis $\bigcup_n\mathcal B_{U,n}$ , then $X$ has a $\sigma$ -locally-finite open basis.

Facts & Assumptions

Given: A locally finite open cover $\mathcal U$ and the stated relative bases.

[L1]

A locally finite family has a neighbourhood at each point meeting only finitely many members (Refinements, locally finite families, point-finite families, and star refinements).

[L2]

A subspace-open set is the intersection of the subspace with an ambient open set (Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace).

Proof

technique · direct

1.1

Since every $U\in\mathcal U$ is open in $X$ , every member of a relative open basis $\mathcal B_{U,n}$ is open in $X$ by [L2]. Put $\mathcal B_n=\bigcup_{U\in\mathcal U}\mathcal B_{U,n}$ .

L2construct

2.1

The family $\mathcal B_n$ is locally finite. At $x$ , take from [L1] a neighbourhood meeting only finitely many $U$ ; within each of those finitely many $U$ , local finiteness of $\mathcal B_{U,n}$ supplies a neighbourhood meeting finitely many members, and their finite intersection meets only finitely many members of $\mathcal B_n$ .

L1step 1.1

2.2

If $O$ is open and $x\in O$ , choose $U\in\mathcal U$ containing $x$ and then a member of the basis of $U$ containing $x$ and contained in $O\cap U$ . Thus $\bigcup_n\mathcal B_n$ is a basis of $X$ .

step 1.1

3.1

Steps 2.1 and 2.2 prove the result.

step 2.1step 2.2∎

Depends on

Used by

Under choice, a space is metrizable if and only if it is paracompact, Hausdorff, and locally metrizable Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 22 results over 11 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

UCR, Partitions of Unity and a Metrization Theorem of Smirnov (standard reference, not scraped)

A locally finite open cover by subspaces with σ\sigmaσ-locally-finite bases yields a σ\sigmaσ-locally-finite basis of the whole space