CorollaryStatement: AI-adaptedProof: AI-generatedprecheck 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 ${A C}_{ω}$ and $D C$ , 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 $R^{n}$ with $n \geq 1$

Statement

Assume ${A C}_{ω}$ and $D C$ . If $n \geq 1$ and $A \subseteq R^{n}$ is nonempty, then the following conditions are equivalent: compactness; closedness and boundedness; pseudocompactness; attainment of a maximum and minimum by every continuous $A \to R$ ; countable compactness; sequential compactness; limit point compactness; and completeness together with total boundedness.

Facts & Assumptions

Given: ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )), $D C$ (The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain), an integer $n \geq 1$ , 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 $R^{n}$ with $n \geq 1$ , compactness, closedness and boundedness, pseudocompactness, and attainment of extrema by every continuous real-valued function are equivalent).

[L2]

Under ${A C}_{ω}$ and $D C$ , 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 $σ$ -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 · two levels

39 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

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

Assuming ACωACω​ and DCDC, 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 RnRn with n≥1n≥1