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 countable choice, is first countable and countably compact but is not separable or Lindelöf
Example
Assume countable choice. Successor ordinals and are isolated. If is a nonzero limit, enumerate the at most countable ordinal and take the successive finite suprema of that enumeration; the result is a countable cofinal sequence, and the corresponding final intervals form a local base at . Thus is first countable. The published ordinal theorem makes it countably compact.
Every at most countable subset is bounded by some , so the nonempty open tail above misses ; hence is not separable. The open initial segments cover , but any at most countable subfamily has bounded union and therefore fails to cover. Thus is not Lindelöf.
Depends on
- First countable space: a countable neighbourhood base at every point
- Separability: the existence of an at most countable dense subset
- Countably compact, Lindel\"of, sequentially compact, limit point compact and $\sigma$-compact spaces, and relatively compact subsets
- The order topology on an ordinal, with the half-open intervals $(\alpha, \beta]$ and the initial segments $[0, \beta]$ as a basis
- Assuming countable choice: every at most countable subset of $\omega_1$ is bounded below $\omega_1$, so no at most countable subset of $\omega_1$ is cofinal in it, and a supremum of at most countably many at most countable ordinals is at most countable
- 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
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
43 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
- UCR General Topology Notes (standard reference, not scraped)
- First uncountable ordinal (Wikipedia) (standard reference, not scraped)
- Order topology (Wikipedia) (standard reference, not scraped)