CorollaryStatement: Literature-sourcedProof: AI-generatedverified 2026-09-26 (gpt-6-sol)

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.

A continuous map from a closed subset extends smoothly exactly when it has a continuous extension and is smooth near the subset

Statement

Assume countable choice. Let $A \subseteq M$ be closed and let $f : A \to N$ be continuous. Then $f$ extends to a smooth map $M \to N$ if and only if it has a continuous extension to $M$ that is smooth on a neighbourhood of $A$ .

Facts & Assumptions

Given: Countable choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )), a closed subset $A \subseteq M$ , and a continuous map $f : A \to N$ .

[L1]

Relative Whitney approximation for manifold-valued maps smooths a continuous extension without changing it near the closed set (Relative Whitney approximation for manifold-valued maps).

Proof

technique · direct

1.1given

If $f$ has a smooth extension $F : M \to N$ , then that extension is in particular continuous and smooth near $A$ .

1.2L1given

Conversely, suppose $F : M \to N$ is continuous, extends $f$ , and is smooth on a neighbourhood of $A$ . Apply [L1] to $F$ and the closed set $A$ . The resulting smooth map $\tilde{F}$ agrees with $F$ on a neighbourhood of $A$ , hence extends $f$ .

2.1step 1.1step 1.2∎

The two implications from steps 1.1 and 1.2 prove the equivalence.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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., Smooth Approximation of Maps Between Manifolds (standard reference, not scraped)