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Claim in a Differential Topology lecture:

Consider the collection $G$ of maps from $\mathbb R→\mathbb R^2$ which intersect the $x$ axis transversally at a single point. Then $G$ is a stable class.

My Problem: Choose a curve that passes through origin, and perturbe it so that it becomes graph of $x^3$ so since $x^3$ doesnot intersect transversally with $x$ axis, so is not $G$ unstable class?

Jale'de jaled
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  • That is not a small perturbation at all. Stability doesn't say that all perturbations remain in the class. – Ted Shifrin Dec 05 '20 at 23:46
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    why not?, if we are given a curve passing through origin, why cant I perturbe a little to make it look like $x^3$ locally? – Micheal Brain Hurts Dec 06 '20 at 17:56
  • Because changing the tangent at $0$ from non-horizontal to horizontal is a big change. Why don't you write down (or study) the definition of stable class. – Ted Shifrin Dec 07 '20 at 03:56

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