DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 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 maps, evenly covered neighbourhoods, fibres, sheets, and trivial coverings

Definition

A covering map is a continuous surjection $p : E \to B$ such that every $b \in B$ has an open neighbourhood $U$ for which $p^{- 1} (U)$ is a disjoint union of open sets $V_{j}$ , called sheets, and each restriction $p ∣_{V_{j}} : V_{j} \to U$ is a homeomorphism (Continuity of a map of topological spaces at a point and globally, Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological, The disjoint union (coproduct) $⨆_{i} X_{i}$ with the final topology of the canonical injections: a set is open exactly when each of its traces is). Such a $U$ is evenly covered, and $p^{- 1} (b)$ is the fibre over $b$ . A covering is trivial when it is isomorphic over $B$ to a product projection $B \times F \to B$ with $F$ discrete.

Depends on

Dependency tree · 2 levels

Sources