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, refuted: every regular space is normal
Statement
Assuming the Axiom of Countable Choice, every regular space is normal.
Facts & Assumptions
Given: The separation-axiom conventions and the deleted Tychonoff plank.
Regularity separates points from disjoint closed sets, whereas normality separates disjoint closed sets from one another (Regular spaces and spaces, with the source disagreement over whether regularity includes stated explicitly, Normal spaces and spaces, with the source disagreement over whether normality includes stated explicitly).
Under countable choice, the deleted Tychonoff plank is regular and not normal (Assuming countable choice, the deleted Tychonoff plank is a regular nonnormal open subspace of a compact Hausdorff normal space).
Refutation
Let be the deleted Tychonoff plank. It is regular by [L1].
The same space is not normal by [L1], contradicting the displayed universal claim.
Depends on
- Assuming countable choice, the deleted Tychonoff plank is a regular nonnormal open subspace of a compact Hausdorff normal space
- Regular spaces and $T_3$ spaces, with the source disagreement over whether regularity includes $T_1$ stated explicitly
- Normal spaces and $T_4$ spaces, with the source disagreement over whether normality includes $T_1$ stated explicitly
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: 82 results over 18 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
- L. A. Steen and J. A. Seebach, Counterexamples in Topology, deleted Tychonoff plank (standard reference, not scraped)
- Tychonoff plank (Wikipedia) (standard reference, not scraped)
- Sierpinski space (Wikipedia) (standard reference, not scraped)