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.

The based path-class model and basic sets for a universal cover

Definition

Fix a path-connected space $X$ and $x_{0} \in X$ . Let $\tilde{X}$ be the set of endpoint-fixed homotopy classes $[α]$ of paths beginning at $x_{0}$ , and put $p ([α]) = α (1)$ . If $U$ is an open path-connected neighbourhood of $α (1)$ on which the inclusion-induced fundamental-group map is trivial, define $B ([α], U)$ to consist of the classes $[α * γ]$ with $γ$ a path in $U$ beginning at $α (1)$ . These sets are the proposed basic neighbourhoods for the path-class model (Homotopies of continuous maps, homotopies relative to a subspace, and path homotopies relative to the endpoints, Semilocally simply connected spaces with explicit basepoint convention, Paths, path-connected spaces and path components).

Depends on

Dependency tree · 2 levels

Sources