6
$\begingroup$

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!

$\endgroup$
4
  • 1
    $\begingroup$ It depends on your choice of topology. $\endgroup$
    – Fly by Night
    May 3, 2016 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, 2016 at 18:56
  • $\begingroup$ @FlybyNight, to me, a submersion is a map whose derivative is surjective. $\endgroup$
    – L.F. Cavenaghi
    May 3, 2016 at 19:41
  • $\begingroup$ @GrumpyParsnip So, differentiability only makes sense with respect to the "standard topologies"? $\endgroup$
    – Fly by Night
    May 4, 2016 at 20:15

1 Answer 1

Reset to default
8
$\begingroup$

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

$\endgroup$
1
  • 1
    $\begingroup$ this is a good one! I do appreciate. $\endgroup$
    – L.F. Cavenaghi
    May 3, 2016 at 19:42

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.