TheoremStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31

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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.

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Assuming countable choice, normality is not hereditary, even to open regular subspaces

Statement

Assuming the Axiom of Countable Choice, normality is not hereditary, even when the subspace is open and regular.

Facts & Assumptions

Given: The deleted Tychonoff plank construction.

[L1]

Under countable choice, its parent $P$ is normal and its deleted-corner subspace $T$ is open, regular, and nonnormal (Assuming countable choice, the deleted Tychonoff plank is a regular nonnormal open subspace of a compact Hausdorff normal space).

[F1]

A hereditary property passes from every space having it to every subspace (Hereditary, open-hereditary and closed-hereditary properties of topological spaces).

Proof

technique · direct

1.1

The parent $P$ in [L1] is normal, but its open regular subspace $T$ is not normal.

2.1

This one normal space and one nonnormal subspace refute the defining universal condition in [F1].

F1step 1.1∎

Depends on

Used by

Preservation ledger for the separation axioms, with T₁ conventions kept explicit Remark

Dependency tree · two levels

16 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)