Alphabeta Math
ExampleConstruction: AI-adaptedVerification: Not suppliedSession-authored (Fable 5 assisted)judge pass (z-ai/glm-5.2)audited 2026-07-31‡ rests on unproved material (inherited)
How 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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

Assuming choice, the lower-limit line is normal while its square is regular and nonnormal

Example

Assume choice. The half-open intervals of the lower-limit line are clopen, and the line is normal by Assuming countable choice, the lower-limit line is normal. Its square remains regular by Arbitrary products of regular spaces are regular, yet the antidiagonal {(x,−x):x∈R}\{(x,-x):x\in\mathbb R\} is closed discrete while Q2\mathbb Q^2 is dense and countable by The lower-limit plane has a countable dense set and a closed discrete antidiagonal of size ∣R∣|\mathbb{R}|.

Jones's bound turns that profile into nonnormality of the square, as proved in Assuming choice, normality is not productive: the normal lower-limit line has a nonnormal square. This separates the productive behaviour of regularity from the nonproductive behaviour of normality without conflating the two separation conditions.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 100 results over 16 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources