ExampleConstruction: AI-adaptedVerification: Not suppliedjudge pass (z-ai/glm-5.2)audited 2026-07-31

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Assuming countable choice, the deleted Tychonoff plank worked as $T_{3}$ but not normal inside its compact Hausdorff normal parent

Example

Assume countable choice. In $P = (ω_{1} + 1) \times (ω + 1)$ remove the corner $(ω_{1}, ω)$ . The resulting plank is regular and $T_{1}$ , hence $T_{3}$ , but it contains the closed edge sets ${ω_{1}} \times ω$ and $ω_{1} \times {ω}$ that have no disjoint open neighbourhoods.

The parent is compact Hausdorff and normal. The obstruction is concrete: a supposed separation gives an ordinal bound for the countably indexed vertical-tail neighbourhoods, and a horizontal-tail neighbourhood beyond that bound meets a vertical-tail neighbourhood. All stated properties, including the location of countable choice, are proved in Assuming countable choice, the deleted Tychonoff plank is a regular nonnormal open subspace of a compact Hausdorff normal space.

Depends on

Assuming countable choice, the deleted Tychonoff plank is a regular nonnormal open subspace of a compact Hausdorff normal space

Used by

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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)

Assuming countable choice, the deleted Tychonoff plank worked as T3T3​ but not normal inside its compact Hausdorff normal parent