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.
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- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Assuming countable choice, refuted: every regular space is normal
Statement
Assuming the Axiom of Countable Choice, every regular space is normal.
Facts & Assumptions
Given: The separation-axiom conventions and the deleted Tychonoff plank.
Regularity separates points from disjoint closed sets, whereas normality separates disjoint closed sets from one another (Regular spaces and spaces, with the source disagreement over whether regularity includes stated explicitly, Normal spaces and spaces, with the source disagreement over whether normality includes stated explicitly).
Under countable choice, the deleted Tychonoff plank is regular and not normal (Assuming countable choice, the deleted Tychonoff plank is a regular nonnormal open subspace of a compact Hausdorff normal space).
Refutation
Let be the deleted Tychonoff plank. It is regular by [L1].
The same space is not normal by [L1], contradicting the displayed universal claim.
Depends on
- Assuming countable choice, the deleted Tychonoff plank is a regular nonnormal open subspace of a compact Hausdorff normal space
- Regular spaces and $T_3$ spaces, with the source disagreement over whether regularity includes $T_1$ stated explicitly
- Normal spaces and $T_4$ spaces, with the source disagreement over whether normality includes $T_1$ stated explicitly
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
19 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)
- Sierpinski space (Wikipedia) (standard reference, not scraped)