CorollaryStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31

How statement and proof provenance work

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.

Assuming $\mathrm{AC}_\omega$ and $\mathrm{DC}$ , compactness, sequential compactness, countable compactness, limit point compactness, completeness and total boundedness, pseudocompactness, closedness and boundedness, and the extreme-value property are equivalent for nonempty subsets of $\mathbb{R}^n$ with $n\ge1$

Statement

Assume $\mathrm{AC}_\omega$ and $\mathrm{DC}$ . If $n\ge1$ and $A\subseteq\mathbb{R}^n$ is nonempty, then the following conditions are equivalent: compactness; closedness and boundedness; pseudocompactness; attainment of a maximum and minimum by every continuous $A\to\mathbb{R}$ ; countable compactness; sequential compactness; limit point compactness; and completeness together with total boundedness.

Facts & Assumptions

Given: $\mathrm{AC}_\omega$ (The Axiom of Countable Choice ( $\mathrm{AC}_\omega$ )), $\mathrm{DC}$ (The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$ -indexed chain), an integer $n\ge1$ , and a nonempty Euclidean subset $A$ .

[L1]

The four Euclidean conditions compactness, closedness and boundedness, pseudocompactness, and the extreme-value property are equivalent in ZF (For a nonempty subset of $\mathbb{R}^n$ with $n\ge1$ , compactness, closedness and boundedness, pseudocompactness, and attainment of extrema by every continuous real-valued function are equivalent).

[L2]

Under $\mathrm{AC}_\omega$ and $\mathrm{DC}$ , a metric space is compact if and only if it is countably compact, limit point compact, sequentially compact, or complete and totally bounded (For a metric space, compact, countably compact, limit point compact, sequentially compact, and complete together with totally bounded are all equivalent, given countable choice and dependent choice).

[L3]

The topological and metric readings of compactness for a Euclidean subspace agree (For a metric space with its metric topology, compactness in the topological sense is compactness in the metric sense, and the two notions of compact subset coincide); the named topological variants have the meanings of Countably compact, Lindel"of, sequentially compact, limit point compact and $\sigma$ -compact spaces, and relatively compact subsets.

Proof

technique · direct

1.1

The ZF part of the chart is exactly [L1].

1.2

In the Euclidean metric, compactness is equivalent to each of countable compactness, limit point compactness, sequential compactness, and completeness together with total boundedness by [L2].

2.1

By [L3], this metric compactness is the compactness already occurring in step 1.1. Joining the two equivalence classes proves the asserted chart.

L3step 1.1step 1.2∎

Depends on

Used by

The Euclidean closed ball and sphere worked through the compactness equivalence chart Example

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 126 results over 21 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

Sequentially compact space (standard reference, not scraped)
Heine-Borel theorem (Wikipedia) (standard reference, not scraped)