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.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.
The converse from Urysohn separation to DMC: a comparison question
Statement
Over ZF, DMC implies Urysohn's lemma (DMC implies Urysohn's lemma). One may ask whether the converse holds. This item states no answer or current research status for that question.
Remarks
A failure of Urysohn's lemma implies failure of DMC by contraposition (A failure of Urysohn separation forces failure of DMC) and also failure of bounded Tietze extension (A Urysohn separation obstruction also obstructs bounded Tietze extension). These conditional implications establish no countermodel existence. Even if a counterexample to Urysohn's lemma were supplied, it would satisfy and would not refute the converse ; that requires a witness of instead.
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Used by
Dependency tree · two levels
20 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
- Eleftherios Tachtsis, The Urysohn Lemma is independent of ZF + Countable Choice (standard reference, not scraped)
- Norbert Brunner, Geordnete Läuchli Kontinuen (standard reference, not scraped)