I am searching for an example of a map $f : \mathbb{R}^2 \to \mathbb{R}$ that is open but is not a submersion... I know that any constant map is not a submersion, but it is indeed closed, I am wondering for an example where $f$ is an open map.

I appreciate any help!

  • 1
    $\begingroup$ It depends on your choice of topology. $\endgroup$ – Fly by Night May 3 '16 at 18:49
  • $\begingroup$ @FlybyNight: as far as I know a "submersion" is only defined for manifolds; hence you need the standard topologies on $\mathbb R$ and $\mathbb R^2$. $\endgroup$ – Cheerful Parsnip May 3 '16 at 18:56
  • $\begingroup$ @FlybyNight, to me, a submersion is a map whose derivative is surjective. $\endgroup$ – L.F. Cavenaghi May 3 '16 at 19:41
  • $\begingroup$ @GrumpyParsnip So, differentiability only makes sense with respect to the "standard topologies"? $\endgroup$ – Fly by Night May 4 '16 at 20:15

Try the map $(x,y) \to x^3.\,\,$

  • $\begingroup$ this is a good one! I do appreciate. $\endgroup$ – L.F. Cavenaghi May 3 '16 at 19:42

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