Alphabeta MathHow statement and proof provenance work
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On a noncompact manifold, this page states Morse genericity as a strong-topology residual theorem
Remark
The compact theorem and the general theorem package different results.
On a compact manifold, On a compact smooth manifold, the Morse functions form an open dense subset in the and hence topology gives an open dense subset in the ordinary or topology. On an arbitrary noncompact manifold, In the strong topology on , the Morse functions form a residual subset is the strong-topology theorem proved on this page.
The drifting-shell counterexample below serves a narrower purpose. It shows that perturbations which are tiny only on each fixed compact set can still create new critical points far out at infinity, so that kind of weak control is not a substitute for the strong topology. By itself, that example does not settle any separate openness claim.
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Sources
- Marco Gualtieri, Topology I: Smooth Manifolds, Part 10 (standard reference, not scraped)
- Shintaro Fushida-Hardy, Morse theory (standard reference, not scraped)