Alphabeta MathHow statement and proof provenance work
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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.
Under countable choice, its parent is normal and its deleted-corner subspace is open, regular, and nonnormal (Assuming countable choice, the deleted Tychonoff plank is a regular nonnormal open subspace of a compact Hausdorff normal space).
A hereditary property passes from every space having it to every subspace (Hereditary, open-hereditary and closed-hereditary properties of topological spaces).
Proof
The parent in [L1] is normal, but its open regular subspace is not normal.
This one normal space and one nonnormal subspace refute the defining universal condition in [F1].
Depends on
Used by
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 76 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)