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.
Implication, preservation, counterexample, and choice ledger for the countability axioms
The implication chain is second countable first countable and, with , second countable separable and Lindelöf. Separable spaces are ccc. The displayed counterexamples show that the reverse implications and the stated hereditary and productive extensions fail.
The cardinal functions use raw finite values, so the metric theorem is , not a blanket equality of all five functions. Countable choice is stated where it is spent: selecting countably many bases or cover subfamilies, taking countable unions of countable sets, and deriving the metric and countable-product equivalences.
Depends on
- Under choice, the five cardinal functions recover first countability, second countability, separability, Lindelöfness, and ccc at the $\aleph_0$ threshold
- Every second countable space is first countable
- Assuming countable choice, every second countable space is separable
- Assuming countable choice, every second countable space is Lindelöf
- Assuming countable choice, a metrizable space is second countable if and only if it is separable if and only if it is Lindelöf
- Second countability is hereditary
- Assuming countable choice, a countable product of second countable spaces is second countable
- Every separable space satisfies the countable chain condition
- Assuming choice, refuted: every ccc space is separable
- Refuted: every first countable space is second countable
- Refuted: every separable space is second countable
- Refuted: separability is hereditary
- Refuted: Lindelöfness is hereditary
- Assuming countable choice, refuted: Lindelöfness is productive
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
47 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)