CorollaryStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck 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.

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$\mathbb{R}^n$ is locally compact and $\sigma$ -compact

Statement

For $n\ge1$ , Euclidean space $\mathbb{R}^n$ is locally compact and $\sigma$ -compact.

Facts & Assumptions

Given: $\mathbb{R}^n$ with $n\ge1$ , its origin $0$ , and its Euclidean norm.

[L1]

Every Euclidean closed ball of positive radius is compact (For $n\ge1$ , every Euclidean closed ball and every Euclidean sphere of positive radius is compact).

[L2]

A space is locally compact when every point has a compact neighbourhood, and it is $\sigma$ -compact when it is a countable union of compact subsets (Locally compact topological space: every point has a compact neighbourhood; and what this says in a metric space, Countably compact, Lindel"of, sequentially compact, limit point compact and $\sigma$ -compact spaces, and relatively compact subsets).

[L3]

For every real $M$ there is a natural $k\ge1$ with $M<k\cdot1_{\mathbb R}$ (Every complete ordered field is Archimedean).

Proof

technique · constructive

1.1

For $x\in\mathbb{R}^n$ , the closed ball $\overline B_2(x,1)$ is compact by [L1] and contains the open ball $B_2(x,1)$ about $x$ . Thus $x$ has a compact neighbourhood.

L1L2

1.2

For each $k\in\mathbb N$ put $K_k:=\overline B_2(0,k+1)$ . Every $K_k$ is compact by [L1].

L1construct

2.1

If $x\in\mathbb R^n$ , [L3] gives $k\ge1$ with $\lVert x\rVert_2<k$ , so $x\in K_k$ . Hence $\mathbb R^n=\bigcup_{k\in\mathbb N}K_k$ .

L3step 1.2choose

3.1

Step 1.1 gives local compactness and step 2.1 gives $\sigma$ -compactness.

L2step 1.1step 2.1discharge-construct∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 110 results over 18 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

Sigma-compact space (standard reference, not scraped)
Locally compact space (Wikipedia) (standard reference, not scraped)

Rn\mathbb{R}^nRn is locally compact and σ\sigmaσ-compact