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.
- 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 choice, the lower-limit line is normal while its square is regular and nonnormal
Example
Assume choice. The half-open intervals of the lower-limit line are clopen, and the line is normal by Assuming countable choice, the lower-limit line is normal. Its square remains regular by Arbitrary products of regular spaces are regular, yet the antidiagonal is closed discrete while is dense and countable by The lower-limit plane has a countable dense set and a closed discrete antidiagonal of size .
Jones's bound turns that profile into nonnormality of the square, as proved in Assuming choice, normality is not productive: the normal lower-limit line has a nonnormal square. This separates the productive behaviour of regularity from the nonproductive behaviour of normality without conflating the two separation conditions.
Depends on
- Assuming countable choice, the lower-limit line is normal
- Arbitrary products of regular spaces are regular
- The lower-limit plane has a countable dense set and a closed discrete antidiagonal of size $|\mathbb{R}|$
- Assuming choice, normality is not productive: the normal lower-limit line has a nonnormal square
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
23 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
- G. Gruenhage, General Topology Course Notes, Sorgenfrey plane (standard reference, not scraped)
- Sorgenfrey topology (Encyclopedia of Mathematics) (standard reference, not scraped)
- Sorgenfrey plane (Wikipedia) (standard reference, not scraped)