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, the lower-limit plane is first countable, separable, and ccc, but not second countable or Lindelöf
Example
Let be the lower-limit line. Under Choice, the countable-product theorem makes first countable, and the rational grid is a countable dense subset; hence is separable and therefore ccc. The antidiagonal is an uncountable closed discrete subspace, as proved in Refuted: separability is hereditary and Assuming countable choice, refuted: Lindelöfness is productive. If were second countable, hereditary second countability would make the discrete space second countable, which is impossible because every basis of a discrete space contains all its singletons. The explicit open cover in Assuming countable choice, refuted: Lindelöfness is productive shows directly that is not Lindelöf.
Depends on
- For the lower-limit line, $\chi=d=L=c=\aleph_0$ and $w=2^{\aleph_0}$ under choice
- Assuming countable choice, a countable product of first countable spaces is first countable
- Every separable space satisfies the countable chain condition
- Second countability is hereditary
- Refuted: separability is hereditary
- Assuming countable choice, refuted: Lindelöfness is productive
- Second countability: an at most countable basis for the topology
- Countably compact, Lindel\"of, sequentially compact, limit point compact and $\sigma$-compact spaces, and relatively compact subsets
- $\mathbb{R}$ is uncountable (Cantor's nested intervals, 1874)
Used by
Dependency tree · two levels
58 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)
- Sorgenfrey plane (Wikipedia) (standard reference, not scraped)