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.
Assuming choice, is countably compact, noncompact, and not paracompact
Example
Assume the Axiom of Choice. The ordinal space is Hausdorff, countably compact, and noncompact by Every successor ordinal is compact in its order topology and every limit ordinal is not; and, assuming countable choice, is countably compact and sequentially compact while is compact and Every ordinal with its order topology has a basis of clopen sets, and is , Hausdorff and regular. If it were paracompact, then Assuming countable choice, every countably compact paracompact Hausdorff space is compact would make it compact. It is therefore not paracompact.
The use of Choice includes the countable choice hypothesis of the cited ordinal compactness result.
Depends on
- Assuming countable choice, every countably compact paracompact Hausdorff space is compact
- Every successor ordinal is compact in its order topology and every limit ordinal is not; and, assuming countable choice, $\omega_1$ is countably compact and sequentially compact while $\omega_1 + 1$ is compact
- Every ordinal with its order topology has a basis of clopen sets, and is $T_1$, Hausdorff and regular
- The Axiom of Choice
Used by
Dependency tree · two levels
40 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
- First uncountable ordinal (Wikipedia) (standard reference, not scraped)
- G. Gruenhage, General Topology Course Notes (standard reference, not scraped)