LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Locally finite families remain locally finite after taking closures, closure commutes with their union, and a locally finite union of closed sets is closed

Statement

Let ${A_{i}}_{i \in I}$ be a locally finite family of subsets of a topological space $X$ . Then ${\overline{A_{i}}}_{i \in I}$ is locally finite and $\overline{⋃_{i \in I} A_{i}} = ⋃_{i \in I} \overline{A_{i}} .$ Consequently, a locally finite union of closed subsets of $X$ is closed.

Facts & Assumptions

Given: A locally finite family ${A_{i}}_{i \in I}$ in a topological space $X$ .

[F1]

Local finiteness says that each point has a neighbourhood meeting only finitely many $A_{i}$ (Refinements, locally finite families, point-finite families, and star refinements).

[L1]

A point belongs to $\overline{A}$ exactly when every neighbourhood of it meets $A$ , and $\overline{A}$ is the smallest closed superset of $A$ (A point lies in the closure of $A$ iff every basic neighbourhood of it meets $A$ ; the closure is the smallest closed superset and equals $A$ together with its derived set).

Proof

technique · direct

1.1

Fix $x \in X$ and a neighbourhood $N$ of $x$ meeting only $A_{i_{1}}, \dots, A_{i_{n}}$ . Choose an open neighbourhood $O$ of $x$ with $O \subseteq N$ . If $O \cap \overline{A_{j}} \neq \emptyset$ , choose $y \in O \cap \overline{A_{j}}$ ; the open neighbourhood $O$ of $y$ then meets $A_{j}$ , so $N$ meets $A_{j}$ and $j \in {i_{1}, \dots, i_{n}}$ .

F1L1

1.2

The inclusion $⋃_{i} \overline{A_{i}} \subseteq \overline{⋃_{i} A_{i}}$ holds because each $\overline{A_{i}}$ is contained in every closed set containing $A_{i}$ , in particular in $\overline{⋃_{i} A_{i}}$ .

2.1

Thus $O$ meets only $\overline{A_{i_{1}}}, \dots, \overline{A_{i_{n}}}$ , so the closed family is locally finite.

step 1.1F1

2.2

Let $x \in \overline{⋃_{i} A_{i}}$ and take $N$ as in step 1.1; if $x \notin ⋃_{i} \overline{A_{i}}$ , then for each $i_{k}$ an open neighbourhood of $x$ misses $A_{i_{k}}$ , and its finite intersection with an open neighbourhood inside $N$ misses every $A_{i}$ , contradicting the closure criterion.

step 1.1L1

3.1

Hence $\overline{⋃_{i} A_{i}} = ⋃_{i} \overline{A_{i}}$ by steps 1.2 and 2.2; if every $A_{i}$ is closed, the right-hand side is $⋃_{i} A_{i}$ , so that union is closed.

step 1.2step 2.2L1∎

Depends on

Used by

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. Robbin, Partitions of Unity (standard reference, not scraped)
S. Semmes, Topology notes, Sections 5.13–5.14 (Rice University) (standard reference, not scraped)