TheoremStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30

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A smooth Urysohn lemma for a closed set in an open set

Statement

Let $A$ be a closed subset of a smooth manifold $M$ , and let $U \subseteq M$ be open with $A \subseteq U$ . Then there exists a smooth function $f : M \to [0, 1]$ such that $f = 1$ on an open neighbourhood of $A$ and $supp (f) \subseteq U$ .

Facts & Assumptions

Given: A closed set $A \subseteq M$ and an open set $U \subseteq M$ with $A \subseteq U$ .

[L1]

Every open cover of a smooth manifold admits a subordinate smooth partition of unity (Smooth partitions of unity exist on manifolds).

[F1]

In a partition of unity subordinate to an open cover, each support lies in its assigned open set and the functions sum to $1$ pointwise (Smooth partitions of unity subordinate to an open cover).

Proof

technique · direct

1.1

The two open sets $U$ and $M ∖ A$ cover $M$ , so [L1] gives smooth functions $ϕ_{U}, ϕ_{A} : M \to [0, 1]$ subordinate to this cover with $ϕ_{U} + ϕ_{A} = 1$ .

L1givenchoose

2.1

Because $supp (ϕ_{A}) \subseteq M ∖ A$ , the function $ϕ_{A}$ vanishes on an open neighbourhood of $A$ ; hence $ϕ_{U} = 1$ there, and $supp (ϕ_{U}) \subseteq U$ by [F1].

F1step 1.1

3.1

Taking $f : = ϕ_{U}$ yields the required smooth function.

step 2.1∎

Depends on

Used by

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

John M. Lee, Introduction to Smooth Manifolds (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)