CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-21

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A proper Euclidean local diffeomorphism over a connected target has constant finite fibre cardinality

Statement

Under the hypotheses of A proper Euclidean local diffeomorphism has finite diffeomorphic sheets near every target point, there is a positive natural number $d$ such that every fibre of $f$ is equinumerous with a set of size $d$ (Equinumerous sets, $A \approx B$ and $A ⪯ B$ ).

Facts & Assumptions

Given: A proper regular $C^{1}$ map $f : U \to V$ with nonempty source and connected target $V$ (Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets).

[L1]

Every $y \in V$ has an open neighbourhood whose preimage is a finite disjoint union of open sets, each carried $C^{1}$ -diffeomorphically onto that neighbourhood by $f$ (A proper Euclidean local diffeomorphism has finite diffeomorphic sheets near every target point).

Proof

technique · direct

1.1

Over a neighbourhood supplied by [L1], every fibre meets each sheet exactly once. Sending a fibre point to its unique sheet gives a bijection from every fibre there to the same nonempty finite sheet index set. Thus fibre cardinality is locally constant.

2.1

For each positive natural $d$ , let $V_{d}$ be the set of target points with fibre cardinality $d$ . Step 1.1 makes every $V_{d}$ open, and the $V_{d}$ form a disjoint cover of $V$ . If two were nonempty, one and the union of all the others would disconnect $V$ . Hence exactly one $V_{d}$ is nonempty, and it is all of $V$ .

step 1.1given∎

Depends on

Used by

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Dependency tree · two levels

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Sources

J. M. Lee, Introduction to Smooth Manifolds, Proposition 2.19 (standard reference, not scraped)