DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-21

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.

Proper maps between Euclidean open sets

Definition

Let $U \subseteq R^{m}$ and $V \subseteq R^{n}$ be open, with their Euclidean metric topologies (The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement). A continuous map $f : U \to V$ is proper when $f^{- 1} [K]$ is compact in $U$ for every compact subset $K$ of $V$ . Continuity and compactness have the meanings of Continuity of a map between metric spaces, at a point and globally, in the $ε$ - $δ$ form and Open cover, subcover, compact metric space, and compact subset of a metric space.

Compactness here is intrinsic to the displayed subspaces. In particular, properness concerns compact subsets of $V$ , not merely subsets compact in the ambient space by an unstated convention.

Depends on

Used by

A proper Euclidean local diffeomorphism has finite diffeomorphic sheets near every target point Theorem

Dependency tree · two levels

8 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

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