CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-24

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.

Euclidean submersions are open maps

Statement

Every $C^{1}$ Euclidean submersion is an open map: if $f : U \subseteq R^{m} \to R^{n}$ is a submersion and $O \subseteq U$ is open, then $f [O]$ is open in $R^{n}$ .

Facts & Assumptions

Given: A Euclidean submersion $f$ and an open subset $O$ of its domain.

[L1]

Near every point of a submersion there are coordinate diffeomorphisms in which the map is a coordinate projection (A Euclidean submersion is locally a coordinate projection).

[L2]

Homeomorphisms and coordinate projections are open maps (Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological, A map into a product is continuous iff each of its components is; the projections are continuous and open; and each projection is surjective when every factor is nonempty, which for an infinite index set uses the Axiom of Choice).

Proof

technique · direct

1.1

If $O = \emptyset$ , its image is open. Otherwise fix $y \in f [O]$ and choose $x \in O$ with $f (x) = y$ .

givenchoose

2.1

Shrink the local source neighbourhood from [L1] so that it lies in $O$ . In the local coordinates, [L2] shows that its image contains an open neighbourhood of $y$ lying in $f [O]$ .

step 1.1L1L2

3.1

Every point of $f [O]$ is therefore interior, so $f [O]$ is open.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

18 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

L. W. Tu, An Introduction to Manifolds, Section 11.2 (standard reference, not scraped)
J. M. Lee, Introduction to Smooth Manifolds, Submersion Theorem (standard reference, not scraped)