CorollaryStatement: Literature-sourcedProof: AI-generated

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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.

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Transverse maps are dense in the strong smooth topology

Statement

Assume the Axiom of Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). For a fixed closed embedded submanifold $Z \subseteq N$ , the smooth maps $M \to N$ that are transverse to $Z$ are dense in the strong smooth topology.

Facts & Assumptions

Given: ${A C}_{ω}$ , smooth manifolds $M, N$ and a closed embedded submanifold $Z \subseteq N$ .

[A1]

Countable Choice is The Axiom of Countable Choice ( ${A C}_{ω}$ ), inherited exactly through the transverse-approximation supplier; no additional family is chosen below.

[L1]

Under [A1], every strong neighbourhood of a smooth map contains a transverse map (Strong Whitney approximation by transverse maps).

Proof

technique · direct

1.1A1L1given

Let $f : M \to N$ be smooth and let $U$ be any strong neighbourhood of $f$ . By [L1], $U$ contains a transverse smooth map.

2.1step 1.1∎

Since this holds for every $f$ and every neighbourhood $U$ , the transverse maps are dense.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

25 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

John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Transversality (standard reference, not scraped)