DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-21

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.

Local homeomorphisms

Definition

A continuous map $f : X \to Y$ is a local homeomorphism when for every $x \in X$ there is an open neighbourhood $U$ of $x$ such that $f [U]$ is open in $Y$ and the restriction

$f ∣_{U} : U ⟶ f [U]$

is a homeomorphism (Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological, Neighbourhood of a point and neighbourhood base, with this library's convention that a neighbourhood need not be open).

Surjectivity is not part of this definition. Nor does the definition require one neighbourhood of a target point to be evenly covered by all of its inverse images, so a local homeomorphism need not be a covering map.

Depends on

Used by

Dependency tree · two levels

8 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

J. Peter May, A Concise Course in Algebraic Topology, Chapter 3, Problem 4 (standard reference, not scraped)