PropositionStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 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.

Local path-connectedness lifts and descends along covering maps

Statement

For a covering $p : E \to B$ , the total space $E$ is locally path-connected if and only if the base $B$ is locally path-connected.

Facts & Assumptions

Given: The objects, hypotheses, and choice principles stated above.

[F1]

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 (def-continuous-map-top, def-homeomorphism-and-open-maps, def-disjoint-union-topology). 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. (Covering maps, evenly covered neighbourhoods, fibres, sheets, and trivial coverings).

[F2]

Let $(X, T)$ be a topological space (def-topological-space) and let $x \in X$ . Subsets carry the subspace topology (def-subspace-topology-top); connectedness is def-connected-space and path-connectedness is def-path-connected. $X$ is locally connected at $x$ when for every open $U$ with $x \in U$ there is an open connected $V$ with $x \in V \subseteq U$ , and locally connected when this holds at every point; $X$ is locally path-connected at $x$ when for every open $U$ with $x \in U$ there is an open path-connected $V$ with $x \in V \subseteq U$ , and locally path-connected when this holds at every point. (Locally connected and locally path-connected spaces: a neighbourhood base of open connected, respectively open path-connected, sets at every point).

[F3]

Let $(X, T_{X})$ and $(Y, T_{Y})$ be topological spaces and let $f : X \to Y$ be a function. Continuity is as in def-continuous-map-top, injections, surjections and bijections as in def-injection-surjection-bijection. $f$ is an open map if $f [U]$ is open in $Y$ for every open $U \subseteq X$ , a closed map if $f [F]$ is closed in $Y$ for every closed $F \subseteq X$ , and a homeomorphism if $f$ is a continuous bijection whose inverse $f^{- 1} : Y \to X$ is also continuous; the spaces are homeomorphic when such an $f$ exists. (Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological).

Proof

technique · direct

1.1

Every point has a sheet homeomorphic to an open neighbourhood of its image.

givenF1F3

2.1

Local path-connectedness passes to open subspaces and across homeomorphisms, which proves both directions using surjectivity to choose a point over each basepoint.

step 1.1F1F2F3

3.1

The preceding construction and implications establish the assertion.

step 2.1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 37 results over 12 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)