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.
Covering-space actions by disjoint translates of neighbourhoods
Definition
A left action of a group on a space by homeomorphisms is a covering-space action when every has an open neighbourhood such that for every nonidentity (Left group actions, transitive actions, and faithful actions, Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological). Acting by homeomorphisms means that each map is a homeomorphism of ; the underlying set action alone would not make the translates open. This condition implies freeness (A free group action has no nonidentity element fixing a point) and makes the translates of pairwise disjoint.
Depends on
Used by
- Every subgroup acts on the universal cover with a connected quotient covering that realizes it Lemma
- The quaternion double cover generates the third homotopy group of SO(3) Lemma
- The orbit map of a covering-space action is a covering, with the acting group equal to the deck group when the total space is path-connected Theorem
Dependency tree · two levels
9 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
- Allen Hatcher, Algebraic Topology, §1.3 (standard reference, not scraped)
- J. Peter May, A Concise Course in Algebraic Topology, Ch. 3 (standard reference, not scraped)
- Marco Gualtieri, MAT1300 Week 4 Term 2, §1.6 (standard reference, not scraped)