Alphabeta MathHow 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.
The Euclidean closed ball and sphere worked through the compactness equivalence chart
Example
Assume and , let , , and . The closed ball and sphere are compact (For , every Euclidean closed ball and every Euclidean sphere of positive radius is compact). Therefore each is closed and bounded, pseudocompact, countably compact, sequentially compact, limit point compact, and complete and totally bounded; every continuous real-valued function on either set attains both extrema (Assuming and , 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 with ). The two sets are those of Euclidean spheres and closed balls as subspaces of .
Depends on
- Euclidean spheres and closed balls as subspaces of $\mathbb{R}^n$
- For $n\ge1$, every Euclidean closed ball and every Euclidean sphere of positive radius is compact
- 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 $n\ge1$
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
16 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
- Heine-Borel theorem (standard reference, not scraped)