CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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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.

Every submersion is an open map

Statement

Every smooth submersion $F : M \to N$ is an open map.

Facts & Assumptions

Given: A smooth submersion $F : M \to N$ .

[F1]

An open map sends open sets to open sets (Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological).

[L1]

Around each point of $M$ , a submersion is locally the projection $(u, v) \mapsto u$ (Local normal form for submersions).

[L2]

Products carry the usual product topology, so if $(u_{0}, v_{0})$ lies in an open set of a product, some product neighbourhood of $(u_{0}, v_{0})$ lies inside that open set (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

1.1L1given

Let $W \subseteq M$ be open and let $y \in F (W)$ . Choose $x \in W$ with $F (x) = y$ . By [L1], after shrinking around $x$ and $y$ , the map is identified with a coordinate projection $(u, v) \mapsto u$ .

2.1step 1.1L2

Since $W$ is open and contains $x$ , [L2] gives a product neighbourhood $U_{1} \times U_{2} \subseteq W$ in those coordinates. The projection sends this product neighbourhood onto the open set $U_{1}$ . Therefore $y$ has an open neighbourhood contained in $F (W)$ .

3.1F1step 2.1∎

Because every point of $F (W)$ is interior, $F (W)$ is open. Thus $F$ is an open map in the sense of [F1].

Depends on

Used by

Dimension formula from roots Proposition
Conjugacy of Cartan subalgebras Theorem

Dependency tree · two levels

17 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

Will J. Merry, Differential Geometry, Proposition 6.13 (standard reference, not scraped)
John M. Lee, Introduction to Smooth Manifolds, Submersions (standard reference, not scraped)