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, normality is not open-hereditary: deleting one corner of the Tychonoff plank
Statement refuted
Assuming countable choice, every open subspace of a normal space is normal.
Facts & Assumptions
Given: The deleted Tychonoff plank construction.
Under countable choice, its parent is normal and deletion of its corner produces an open nonnormal subspace (Assuming countable choice, the deleted Tychonoff plank is a regular nonnormal open subspace of a compact Hausdorff normal space).
Counterexample
Take the compact Hausdorff normal plank of [L1] and its deleted-corner subspace .
The subspace is open in and is not normal, which refutes the displayed assertion.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
12 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
- L. A. Steen and J. A. Seebach, Counterexamples in Topology, deleted Tychonoff plank (standard reference, not scraped)
- Tychonoff plank (Wikipedia) (standard reference, not scraped)