I'm studying differential topology, I know that there are many properties which are stable, but I don't know if to be a Morse function is a stable property. I think that it is not true, but I haven't found a counterexample. Thanks!
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Is your manifold compact? – Ted Shifrin May 27 '18 at 15:45
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Yes, thank you Ted. – Alexandre Muñoz May 27 '18 at 17:59
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2Then stability holds. For me the easiest argument is that when $M$ is compact, maps $M\to Y$ that are transverse to any closed submanifold $Z\subset Y$ form a stable family. And $f\colon M\to\Bbb R$ is Morse if and only if $\text{grad}, f$ is transverse to the zero section of $TM$. – Ted Shifrin May 27 '18 at 18:03
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there is a problem in Guillemin and pollack 1.7.18 that proves stability of Morse functions. – Intuition Nov 11 '18 at 08:05