CorollaryStatement: AI-adaptedProof: AI-generatedprecheck 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.

$R^{n}$ is locally compact and $σ$ -compact

Statement

For $n \geq 1$ , Euclidean space $R^{n}$ is locally compact and $σ$ -compact.

Facts & Assumptions

Given: $R^{n}$ with $n \geq 1$ , its origin $0$ , and its Euclidean norm.

[L1]

Every Euclidean closed ball of positive radius is compact (For $n \geq 1$ , 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 $σ$ -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 $σ$ -compact spaces, and relatively compact subsets).

[L3]

For every real $M$ there is a natural $k \geq 1$ with $M < k \cdot 1_{R}$ (Every complete ordered field is Archimedean).

Proof

technique · constructive

1.1

For $x \in R^{n}$ , 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 $k \in N$ put $K_{k} : = {\overline{B}}_{2} (0, k + 1)$ . Every $K_{k}$ is compact by [L1].

L1construct

2.1

If $x \in R^{n}$ , [L3] gives $k \geq 1$ with $∥ x ∥_{2} < k$ , so $x \in K_{k}$ . Hence $R^{n} = ⋃_{k \in N} K_{k}$ .

L3step 1.2choose

3.1

Step 1.1 gives local compactness and step 2.1 gives $σ$ -compactness.

L2step 1.1step 2.1discharge-construct∎

Depends on

Used by

Dependency tree · two levels

29 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

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

RnRn is locally compact and σσ-compact