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.
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
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Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 72 results over 15 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)