PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck 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

1.1givenF1

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

2.1step 1.1F1

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

3.1step 2.1F1

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.

4.1step 3.1F1

The resulting two-level sheets evenly cover the composite.

5.1step 4.1F1

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

6.1step 5.1∎

The preceding construction and implications establish the assertion.

Depends on

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

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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)