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Regular values form a dense $G_{δ}$ set

Statement

Assume ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). For a smooth map $F : M \to N$ , the set of regular values is a dense $G_{δ}$ subset of $N$ .

Facts & Assumptions

Given: ${A C}_{ω}$ and a smooth map $F : M \to N$ .

[L1]

Under ${A C}_{ω}$ , the critical value set is $σ$ -compact (The critical value set of a smooth map is sigma-compact).

[L2]

Regular values are dense, and the critical value set is null (Regular values have null complement and are dense).

Proof

technique · direct

1.1L1given

By [L1], write the critical value set as $⋃_{j \geq 1} K_{j}$ with each $K_{j}$ compact.

2.1L2step 1.1casesalgebra

If $\dim N > 0$ , then [L2] makes each $K_{j}$ a compact null set, hence it has empty interior. Therefore $N ∖ K_{j}$ is open and dense. The regular-value set is $N ∖ ⋃_{j \geq 1} K_{j} = ⋂_{j \geq 1} (N ∖ K_{j}),$ so it is a dense $G_{δ}$ . If $\dim N = 0$ , then [L2] already says every value is regular, so the regular-value set is all of $N$ , again a dense $G_{δ}$ .

3.1step 2.1∎

Therefore regular values form a dense $G_{δ}$ set.

Depends on

Used by

Dependency tree · two levels

14 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

Marco Gualtieri, Topology I: Smooth Manifolds, cumulative notes (standard reference, not scraped)

Regular values form a dense GδGδ​ set