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.

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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

Used by

For a nonempty path-connected locally path-connected semilocally simply connected space, the path-class projection is a covering map Lemma

Dependency tree · two levels

20 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)