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.

A composite of covering maps is a covering when the outer covering is finite-sheeted

Statement

If $p : E \to B$ and $q : B \to X$ are covering maps and $q$ is finite-sheeted, then $q \circ p : E \to X$ is a covering map.

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).

Proof

technique · direct

1.1

For coverings $p : E \to B$ and $q : B \to X$ , evenly cover a neighbourhood of $x$ for $q$ .

givenF1

2.1

There are only finitely many resulting $q$ -sheets.

step 1.1F1

3.1

Around the unique point of each such sheet over $x$ , choose a smaller neighbourhood evenly covered by $p$ ; intersect their finitely many images downstairs and restrict all sheets to that common neighbourhood.

step 2.1F1

4.1

The resulting two-level sheets evenly cover the composite.

step 3.1F1

5.1

Separate the empty base case, and record that the finite-sheet hypothesis is what permits the common intersection.

step 4.1F1

6.1

The preceding construction and implications establish the assertion.

step 5.1∎

Depends on

Covering maps, evenly covered neighbourhoods, fibres, sheets, and trivial coverings

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 12 results over 9 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)