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.
FALSE: every compact Hausdorff topological group is profinite
Statement
Every compact Hausdorff topological group is profinite.
Facts & Assumptions
Given: The circle group under complex multiplication.
A topological group is profinite only if it is compact, Hausdorff, and totally disconnected (Assuming Choice, a topological group is profinite exactly when it is compact, Hausdorff, and totally disconnected).
Refutation
The circle group is compact and Hausdorff in its Euclidean subspace topology, and it is a topological group under multiplication.
The circle group is connected, so it is not totally disconnected. Therefore [L1] shows that it is not profinite.
This gives a compact Hausdorff group that is not profinite, so the statement is false.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
14 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
- Brian Osserman, Math 6112 notes on inverse limits and profinite groups (standard reference, not scraped)
- H. W. Lenstra, Profinite groups and Galois groups (standard reference, not scraped)