DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-16

How 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 $G$ on a space $E$ by homeomorphisms is a covering-space action when every $e \in E$ has an open neighbourhood $U$ such that $g U \cap U = \emptyset$ for every nonidentity $g \in G$ (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 $e \mapsto g \cdot e$ is a homeomorphism of $E$ ; the underlying set action alone would not make the translates $g U$ open. This condition implies freeness (A free group action has no nonidentity element fixing a point) and makes the translates of $U$ pairwise disjoint.

Depends on

Used by

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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 28 results over 9 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click 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)